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BkiUdoY4dbghMG6xpZYO | 5 | 1 | \section*{Acknowledgements}
We wish to thank Dr. Andrea Geralico and Dr. Damiano Tommasini for the support with the numerical analysis and Prof. Salvatore Capozziello for useful discussions. This work was partially supported by DGAPA-UNAM, Grant No. 113514, and CONACyT, Grant No. 166391.
| train/arxiv |
BkiUdoU5qsBC8zW3y8KN | 5 | 1 | \section{Introduction}
\begin{figure}
\centering
\includegraphics[trim={0.2cm 0cm 0.4cm 0cm},clip,width=0.3\textwidth]{freq_bias.pdf}
\vspace{-2mm}
\caption{Typical popularity distribution of items depicting the long tail.\label{fig:freq_bias}}
\vspace{-4mm}
\end{figure}
The goal of session-based recommendation (SR) models is to recommend top-K items to a user based on the sequence of items clicked so far.
Recently, several effective models for SR based on deep neural networks architectures have been proposed \cite{wu2018session,liu2018stamp,li2017neural,wang2019collaborative}.
These approaches consider SR as a multi-class classification problem where input is a sequence of items clicked in the past in a session and target classes correspond to available items in the catalog.
Many of these approaches use sequential models like recurrent neural networks considering a session as a sequence of item click events \cite{hidasi2015session,jannach2017recurrent,wang2019collaborative}.
On the other hand, approaches like STAMP \cite{liu2018stamp} consider a session to be a set of items, and use attention models while learning to weigh (attend to) items as per their relevance to predict the next item.
Approaches like NARM \cite{li2017neural} and CSRM \cite{wang2019collaborative} use a combination of sequential and attention models.
An important building block in most of these deep learning approaches is their ability to learn representations or embeddings for items and sessions.
Recently, an alternative approach, namely SR-GNN \cite{wu2018session}, has been proposed to model the sessions as graph-structured data using GNNs \cite{li2015gated} rather than as sequences or sets, noting that users tend to make complex \textit{to-and-fro} transitions across items within a session:
for example, consider a session $s = i_1,i_2,i_1,i_3,i_4$ of item clicks by a user. Here, the user clicks on item $i_1$, then clicks on item $i_2$ and then re-clicks on item $i_1$.
This sequence of clicks induces a graph where nodes and edges correspond to items and transitions across items, respectively.
For session $s$ in the above example, the fact that $i_2$ and $i_3$ are neighbors of $i_1$ in the induced session-graph, the representation of $i_1$ can be updated using representations of its neighbors, i.e. $i_2$ and $i_3$, and thus obtain a more context-aware and informative representations.
It is worth noting that this way of capturing neighborhood information has also been found to be effective in neighborhood-based SR methods such as SKNN \cite{jannach2017recurrent} and STAN \cite{garg2019sequence}.
It is well-known that more popular items are presented and interacted-with more often on online platforms. This results in a skewed distribution of items clicked by users \cite{steck2011item,abdollahpouri2017controlling,yang2018unbiased}, as illustrated in Fig. \ref{fig:freq_bias}.
The models trained using the resulting data tend to have \textit{popularity bias}, i.e. they tend to recommend more popular items over rarely clicked items.
We note that SR-GNN (referred to as GNN hereafter) also suffers from popularity bias.
This problem is even more severe in a practical online setting where new items are frequently added to the catalog, and are inherently less popular during initial days.
To mitigate this problem, we study GNN through the lens of an \textit{item and session-graph representation learning mechanism}, where the goal is to obtain a session-graph representation that is \textit{similar} to the representation of the item likely to be clicked next.
We motivate the advantage of restricting the item and session-graph representations to lie on a unit hypersphere both during training and inference, and propose \textbf{NISER}: \textbf{N}ormalized \textbf{I}tem and \textbf{Se}ssion \textbf{R}epresentations model for SR.
We demonstrate the enhanced ability of NISER to deal with popularity bias in comparison to a vanilla GNN model in the offline as well as online settings.
We also extend NISER to incorporate the sequential nature of a session via position embeddings \cite{vaswani2017attention}, thereby leveraging the benefits of both sequence-aware models (like RNNs) and graph-aware models.
\section{Related Work}
Recent results in computer vision literature, e.g. \cite{wang2017normface,zheng2018ring}, indicate the effectiveness of normalizing the final image features during training, and argue in favor of cosine similarity over inner product for learning and comparing feature vectors.
\cite{zheng2018ring} introduces the ring loss for soft feature normalization which eventually learns to constrain the feature vectors on a unit hypersphere. Normalizing words embeddings is also popular in NLP applications, e.g. \cite{peng2015comparative} proposes penalizing the L$_2$ norm of word embeddings for regularization.
However, to the best of our knowledge, the idea of normalizing item and session-graph embeddings or representations has not been explored.
In this work, we study the effect of normalizing the embeddings on popularity bias which has not been established and studied so far.
Several approaches to deal with popularity bias exist in the collaborative filtering settings, e.g. \cite{abdollahpouri2017controlling,steck2011item,yang2018unbiased}.
To deal with popularity bias, \cite{abdollahpouri2017controlling} introduces the notion of flexible regularization in a learning-to-rank algorithm. Similarly \cite{steck2011item,yang2018unbiased} uses the power-law of popularity where the probability of recommending an item is a smooth function of the items' popularity, controlled by an exponent factor. However, to the best of our knowledge, this is the first attempt to study and address popularity bias in DNN-based SR using SR-GNN \cite{wu2018session} as a working example.
Furthermore, SR-GNN does not incorporate the sequential information explicitly to obtain the session-graph representation. We study the effect of incorporating position embeddings \cite{vaswani2017attention} and show that it leads to minor but consistent improvement in recommendation performance.
\section{Problem Definition\label{sec:prob_def}}
Let $\mathcal{S}$ denote all past sessions, and $\mathcal{I}$ denote the set of $m$ items observed in the set $\mathcal{S}$.
Any session $s \in \mathcal{S}$ is a chronologically ordered tuple of item-click events:
$s = (i_{s,1},i_{s,2},\ldots,i_{s,l})$, where each of the $l$ item-click events $i_{s,j}$ ($j=1\ldots l$) corresponds to an item in $\mathcal{I}$, and $j$ denotes the position of the item $i_{s,j}$ in the session $s$.
A session $s$ can be modeled as a graph $\mathcal{G}_s = (\mathcal{V}_s, \mathcal{E}_s)$, where $i_{s,j}\in \mathcal{V}_s$ is a node in the graph.
Further, $(i_{s,j},i_{s,j+1})\in \mathcal{E}_s$ is a directed edge from $i_{s,j}$ to $i_{s,j+1}$.
Given $s$, the goal of SR is to predict the next item $i_{s,l+1}$ by estimating the $m$-dimensional probability vector $\mathbf{\hat{y}}_{s,l+1}$ corresponding to the relevance scores for the $m$ items.
The $K$ items with highest scores constitute the top-K recommendation list.
\section{Learning Item and Session Representations\label{sec:approach}}
Each item is mapped to a $d$-dimensional vector from the trainable embedding look-up table or matrix $\mathbf{I} = [\mathbf{i}_1,\mathbf{i}_2,\ldots,\mathbf{i}_m]^T \in \mathbb{R}^{m\times d}$ such that each row $\mathbf{i}_j\in \mathbb{R}^d$ is the $d$-dimensional representation or embedding\footnote{We use the terms representation and embedding interchangeably.} vector corresponding to item $i_j \in \mathcal{I}$ ($j=1\ldots m$).
Consider any function $f$ (e.g. a neural network as in \cite{liu2018stamp,wu2018session})---parameterized by $\boldsymbol\theta$---that maps the items in a session $s$ to session embedding $\mathbf{s} = f(\mathbf{I}_s;\boldsymbol\theta)$, where\footnote{To ensure same dimensions of $\mathbf{I}_s \in \mathbb{R}^{L\times d}$ across sessions, we can pad with a dummy vector $L-l$ times.} $\mathbf{I}_s = [\mathbf{i}_{s,1},\mathbf{i}_{s,2},\ldots,\mathbf{i}_{s,l}]^T \in \mathbb{R}^{l\times d}$.
Along with $\mathbf{I}_s$ as an input which considers $s$ as a sequence, we also introduce an adjacency matrix $\mathbf{A}_s$ to incorporate the graph structure. We discuss this in more detail later in Section \ref{ssec:niser-gnn}.
The goal is to obtain $\mathbf{s}$ that is close to the embedding $\mathbf{i}_{s,l+1}$ of the target item $i_{k}=i_{s,l+1}$, such that the estimated index/class for the target item is $k = \argmax_j~ \mathbf{i}_j^T\mathbf{s}$ with $j=1\ldots m$.
In a DNN-based model $f$, this is approximated via a differentiable softmax function such that the probability of next item being $i_k$ is given by:
\begin{equation}
\label{eq:softmax1}
p_k(\mathbf{s}) = \hat{\mathbf{y}}_k = \frac{\text{exp}(\mathbf{i}_k^T\mathbf{s})}{\sum_{j=1}^{m}\text{exp}(\mathbf{i}_j^T\mathbf{s})}.
\end{equation}
For this $m$-way classification task, softmax (cross-entropy) loss is used during training for estimating $\boldsymbol \theta$ by minimizing the sum of $\mathcal{L(\hat{\mathbf{y}})} = - \sum_{j=1}^{m}\mathbf{y}_j\text{log}(\hat{\mathbf{y}}_j)$
over all training samples, where $\mathbf{y} \in \{0,1\}^m$ is a 1-hot vector with $\mathbf{y}_k=1$ corresponding to the correct (target) class $k$.
We next introduce the radial property \cite{wang2017normface,zheng2018ring} of softmax loss, and then use it to motivate the need for normalizing the item and session representations in order to reduce popularity bias.
\subsection{Radial Property of Softmax Loss}
\begin{figure*}[h]
\includegraphics[scale=0.5,trim={1cm 4.5cm 2cm 4cm},clip]{nise-15.pdf}
\caption{Illustrative flow diagram for NISER.\label{fig:niser}}
\end{figure*}
It is well-known that optimizing for the softmax loss leads to a radial distribution of features for a target class \cite{wang2017normface,zheng2018ring}:
If $k = \argmax_j~ \mathbf{i}_j^T\mathbf{s}$, then it is easy to show that
\begin{equation}
\frac{\text{exp}(\sigma \mathbf{i}_k^T\mathbf{s})}{\sum_{j\neq k}\text{exp}(\mathbf{i}_j^T\mathbf{s})~+~\text{exp}(\sigma \mathbf{i}_k^T\mathbf{s})} > \frac{\text{exp}(\mathbf{i}_k^T\mathbf{s})}{\sum_{j=1}^{m}\text{exp}(\mathbf{i}_j^T\mathbf{s})}
\end{equation}
for any $\sigma>1$.
This, in turn, implies that softmax loss favors large norm of features for easily classifiable instances.
We omit details for brevity and refer the reader to \cite{wang2017normface,zheng2018ring} for a thorough analysis and proof.
This means that a high value of $\hat{\mathbf{y}}_k$ can be attained by multiplying vector $\mathbf{i}_k$ by a scalar value $\sigma >1$; or simply by ensuring a large norm for the item embedding vector.
\begin{figure}[h]
\centering
\includegraphics[trim={0.25cm 0cm 0.4cm 0cm},clip,width=0.35\textwidth]{pop-avg-norm.pdf}
\vspace{-2mm}
\caption{Recall@20 and L$_2$ norm of learned item embeddings decreases with decreasing popularity in GNN \cite{wu2018session}.\label{fig:pop}}
\vspace{-4mm}
\end{figure}
\subsection{Normalizing the Representations \label{ssec:motiv}}
We note that the radial property has an interesting implication in the context of popularity bias: target items that are easier to predict are likely to have higher L$_2$ norm.
We illustrate this with the help of an example: Items that are popular are likely to be clicked more often, and hence the trained parameters $\boldsymbol{\theta}$ and $\mathbf{I}$ should have values that ensure these items get recommended more often.
The radial property implies that for a given input $s$ and a popular target item $i_k$, a correct classification decision can be obtained as follows: learn the embedding vector $\mathbf{i}_k$ with high $||\mathbf{i}_k||_2$ such that the inner product $\mathbf{i}_k^T\mathbf{s} = ||\mathbf{i}_k||_2||\mathbf{s}||_2cos\alpha$ (where $\alpha$ is the angle between the item and session embeddings) is likely to be high to ensure large value for $\hat{\mathbf{y}}_k$ (even when $\alpha$ is not small enough and $||s||_2$ is not large enough).
When analyzing the item embeddings from a GNN model \cite{wu2018session}, we observe that this is indeed the case: As shown in Fig. \ref{fig:pop}, items with high popularity have high L$_2$ norm while less popular items have significantly lower L$_2$ norm.
Further, performance of GNN (depicted in terms of Recall@20) degrades as the popularity of the target item decreases.
\subsection{NISER}
Based on the above observations, we consider to minimize the influence of embedding norms in the final classification and recommendation decision. We propose optimizing for cosine similarity as a measure of similarity between item and session embeddings instead of the above-stated inner product.
Therefore, during training as well as inference, we normalize the item embeddings as
$\tilde{\mathbf{i}}_k = \frac{\mathbf{i}_{k}}{||\mathbf{i}_{k}||_{_2}}$,
and use them to get $\tilde{\mathbf{I}}_s$.
The session embedding is then obtained as
$\mathbf{s} = f(\tilde{\mathbf{I}}_s;\boldsymbol\theta)$, and is similarly normalized to $\tilde{\mathbf{s}}$ to enforce a unit norm.
The normalized item and session embeddings are then used to obtain the relevance score for next clicked item $i_k$ computed as,
\begin{equation} \label{eq:softmax}
\hat{\mathbf{y}}_k=\frac{\text{exp}(\sigma\tilde{\mathbf{i}}_k^T\tilde{\mathbf{s}})}{\sum_{j=1}^{m}\text{exp}(\sigma \tilde{\mathbf{i}}_j^T\tilde{\mathbf{s}})}.
\end{equation}
Note that the cosine similarity $\tilde{\mathbf{i}}_k^T\tilde{\mathbf{s}}$ is restricted to $[-1,1]$. As shown in \cite{wang2017normface}, this implies that the softmax loss is likely to get saturated at high values for the training set: a scaling factor $\sigma>1$ is useful in practice to allow for better convergence.
\section{Leveraging NISER with GNN\label{ssec:niser-gnn}}
We consider learning the representations of items and session-graphs with GNNs where the session-graph is represented by $\mathcal{G}_s$ as introduced in Section \ref{sec:prob_def}.
Consider two normalized adjacency matrices $\mathbf{A}^{in}_s \in \mathbb{R}^{l,l}$ and $\mathbf{A}^{out}_s \in \mathbb{R}^{l,l}$ corresponding to the incoming and outgoing edges in graph $\mathcal{G}_s$ as illustrated in Fig. \ref{fig:niser}.
GNN takes adjacency matrices $\mathbf{A}^{in}_s$ and $\mathbf{A}^{out}_s$, and the normalized item embeddings $\tilde{\mathbf{I}}_s$ as input, and returns an updated set of embeddings after $\tau$ iterations of message propagation across vertices in the graph using gated recurrent units \cite{li2015gated}: $[\tilde{\mathbf{i}}^\tau_{s,1},\tilde{\mathbf{i}}^\tau_{s,2},\ldots,\tilde{\mathbf{i}}^\tau_{s,l}] = G(\mathbf{A}^{in}_s,\mathbf{A}^{out}_s,\tilde{\mathbf{I}}_s;\boldsymbol{\theta}_g)$, where $\boldsymbol{\theta}_g$ represents the parameters of the GNN function $G$.
For any node in the graph, the current representation of the node and the representations of its neighboring nodes are used to iteratively update the representation of the node $\tau$ times.
More specifically, the representation of node $i_{s,j}$ in the $t$-th message propagation step is updated as follows:
\begin{align}
\label{eq:adj}
\mathcal{\mathbf{a}}_{s,j}^t &= [\mathcal{\mathbf{A}}^{in}_{s,j:} \tilde{\mathbf{I}}^{t-1}_s \mathbf{H}_1, \mathcal{\mathbf{A}}^{out}_{s,j:} \tilde{\mathbf{I}}^{t-1}_s \mathbf{H}_2]^T + \mathbf{b},\\
\mathcal{\mathbf{z}}_{s,j}^t &= \sigma (\mathbf{W}_z \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_z\tilde{\mathbf{i}}_{s,j}^{t-1}),\\
\mathbf{r}_{s,j}^t &= \sigma (\mathbf{W}_r \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_r\tilde{\mathbf{i}}_{s,j}^{t-1}),\\
\hat{\mathbf{i}}_{s,j}^t &= \tanh (\mathbf{W}_o \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_o (\mathbf{r}_{s,j}^t \odot \tilde{\mathbf{i}}_{s,j}^{t-1})),\\
\tilde{\mathbf{i}}_{s,j}^t &= (1-\mathbf{z}_{s,j}^t)\odot \tilde{\mathbf{i}}_{s,j}^{t-1} + \mathbf{z}_{s,j}^t \odot \hat{\mathbf{i}}_{s,j}^t,
\end{align}
\iffalse
\begin{align}
\hat{\mathbf{i}}_{s,j}^t &= \tanh (\mathbf{W}_o \mathcal{\mathbf{a}}_{s,j}^t + \mathbf{U}_o (\mathbf{r}_{s,j}^t \odot \tilde{\mathbf{i}}_{s,j}^{t-1})),\\
\tilde{\mathbf{i}}_{s,j}^t &= (1-\mathbf{z}_{s,j}^t)\odot \tilde{\mathbf{i}}_{s,j}^{t-1} + \mathbf{z}_{s,j}^t \odot \hat{\mathbf{i}}_{s,j}^t,
\end{align}
\fi
where $\mathbf{A}^{in}_{s,j:}$, $\mathbf{A}^{out}_{s,j:} \in \mathbb{R}^{1\times l}$ depicts the $j$-th row of $\mathbf{A}^{in}_{s}$ and $\mathbf{A}^{out}_{s}$ respectively, $\mathbf{H}_1$, $\mathbf{H}_2 \in \mathbb{R}^{d \times d}$, $\mathbf{W}_{(.)}$ and $\mathbf{U}_{(.)}$ are trainable parameters, $\sigma (.)$ is the sigmoid function, and $\odot$ is the element-wise multiplication operator.
To incorporate sequential information of item interactions, we optionally learn \textit{position embeddings} and add them to item embeddings to effectively obtain position-aware item (and subsequently session) embeddings.
The final embeddings for items in a session are computed as $\tilde{\mathbf{i}}^{\tau,p}_{s,j} = \tilde{\mathbf{i}}^\tau_{s,j} + \mathbf{p}_j$, where $\mathbf{p}_j \in \mathbb{R}^d$ is embedding vector for position $j$ obtained via a lookup over the position embeddings matrix $\mathbf{P} = [\mathbf{p}_1,\mathbf{p}_2,\ldots,\mathbf{p}_L]^T \in \mathbb{R}^{L\times d}$, where $L$ denotes the maximum length of any input session such that position $l\leq L$.
The soft-attention weight of the $j$-th item in session $s$ is computed as
$\alpha_j = \mathbf{q}^T\text{sigmoid}(\mathbf{W}_1\tilde{\mathbf{i}}^{\tau,p}_{s,l} + \mathbf{W}_2\tilde{\mathbf{i}}^{\tau,p}_{s,j} + \mathbf{c})$,
where $\mathbf{q}, \mathbf{c}\in \mathbb{R}^d$, $\mathbf{W}_1,\mathbf{W}_2 \in \mathbb{R}^{d\times d}$.
The $\alpha_j$s are further normalized using softmax.
An intermediate session embedding $\mathbf{s}'$ is computed as:
$\mathbf{s}' = \sum_{j=1}^{t}\alpha_j\tilde{\mathbf{i}}^{\tau,p}_{s,j}$.
The session embedding $\mathbf{s}$ is a linear transformation over the concatenation of intermediate session embedding $\mathbf{s}'$ and the embedding of most recent item $\tilde{\mathbf{i}}^{\tau,p}_{s,l}$, s.t. $\mathbf{s} = \mathbf{W}_3[\mathbf{s}';\tilde{\mathbf{i}}^{\tau,p}_{s,l}]
$, where $\mathbf{W}_3\in \mathbb{R}^{d\times2d}$.
The final recommendation scores for the $m$ items are computed as per Eq. \ref{eq:softmax}.
Note that while the session-graph embedding is obtained using item embeddings $\tilde{\mathbf{i}}^{\tau,p}_{s,t}$ that are aware of the session-graph and sequence, the normalized item embeddings $\tilde{\mathbf{i}}_j$ ($j=1\ldots m$) independent of a particular session are used to compute the recommendation scores.
\section{Experimental Evaluation}
\textbf{Dataset Details:}
We evaluate NISER on publicly available benchmark datasets: i) Yoochoose (YC), ii) Diginetica (DN), and iii) RetailRocket (RR).
The YC\footnote{http://2015.recsyschallenge.com/challege.html} dataset is from RecSys Challenge 2015, which contains a stream of user clicks on an e-commerce website within six months.
Given the large number of sessions in YC, the recent 1/4 and 1/64 fractions of the training set are used to form two datasets: YC-1/4 and YC-1/64, respectively, as done in \cite{wu2018session}.
The DN\footnote{http://cikm2016.cs.iupui.edu/cikm-cup} dataset is transactional data from CIKM Cup 2016 challenge.
The RR\footnote{https://www.dropbox.com/sh/dbzmtq4zhzbj5o9/AACldzQWbw-igKjcPTBI6ZPAa?dl=0} dataset is from an e-commerce personalization company retailrocket, which published dataset with six month of user browsing activities.
\\
\textbf{Offline and Online setting:}
We consider two evaluation settings: i. offline and ii. online.
For evaluation in offline setting, we consider static splits of train and test as used in \cite{wu2018session} for YC and DN.
For RR, we consider sessions from last 14 days for testing and remaining 166 days for training.
The statistics of datasets are summarized in Table \ref{tab:datastats}.
For evaluation in online setting, we re-train the models every day for 2 weeks (number of sessions per day for YC is much larger, so we evaluate for 1 week due to computational constraints) by appending the sessions from that day to the previous train set, and report the test results of the trained model on sessions from the subsequent day.\\
\textbf{NISER and its variants:}
We apply our approach over GNN and adapt code\footnote{https://github.com/CRIPAC-DIG/SR-GNN} from \cite{wu2018session} with suitable modification described later.
We found that for sessions with length $l>10$, considering only the most recently clicked 10 items to make recommendations worked consistently better across datasets.
We refer to this variant as GNN+ and use this additional pre-processing step in all our experiments.
\begin{table}[t]
\caption{Statistics of the datasets used for offline experiments. \label{tab:datastats}}
\vspace{-2mm}
\scalebox{0.9}{
\begin{tabular}{|p{0.25\linewidth}|c|c|c|c|}
\hline
\textbf{Statistics} & \textbf{DN} & \textbf{RR} & \textbf{YC-1/64} & \textbf{YC-1/4}\\
\hline
\#train sessions&0.7 M&0.7 M&0.4 M&0.6 M\\
\#test sessions&60,858&60,594&55,898&55,898\\
\#items&43,097&48,759&16,766&29,618\\
Average length&5.12&3.55&6.16&5.17\\
\hline
\end{tabular}}
\vspace{-4mm}
\end{table}
We consider enhancement over GNN+, and proposed following variants of the embedding normalization approach:
\begin{itemize}
\item \textit{Normalized Item Representations (NIR)}: only item embeddings are normalized and scale factor $\sigma$ is not used\footnote{$\tilde{\mathbf{i}}_k^T\mathbf{s}$ is not restricted to [-1,1], in general $||\mathbf{s}||_{_2}\neq1$.},
\item \textit{Normalized Item and Session Representations (NISER)}: both item and session embeddings are normalized,
\item \textit{NISER+}: NISER with position embeddings and dropout applied to input item embeddings.
\end{itemize}
\textbf{Hyperparameter Setup:}
Following \cite{wu2018session},
we use $d=100$ and learning rate of 0.001 with Adam optimizer.
We use 10\% of train set which is closer to test test in time as hold-out validation set for hyperparameter tuning including scale factor $\sigma$.
We found $\sigma = 16.0$ to work best across most models trained on respective hold-out validation set chosen from \{$4.0,9.0,16.0,25.0$\}, and hence, we use the same value across datasets for consistency.
We use dropout probability of 0.1 on dimensions of item embeddings in NISER+ across all models.
Since the validation set is closer to the test set in time, therefore, it is desirable to use it for training the models.
After finding the best epoch
via early stopping based on performance on validation set, we re-train the model for same number of epochs on combined train and validation set.
We train five models for the best hyperparameters with random initialization, and report average and standard deviation of various metrics for all datasets except for YC-1/4 where we train three models (as it is a large dataset and takes around ~15 hours for training one model).
\\
\textbf{Evaluation Metrics:}
We use same evaluation metrics Recall@K and Mean Reciprocal Rank (MRR@K) as in \cite{wu2018session} with $K=20$.
\textbf{Recall@K} represents the proportion of test instances which has desired item in the top-K items.
\textbf{MRR@K} (Mean Reciprocal Rank) is the average of reciprocal ranks of desired item in recommendation list.
Large value of MRR indicates that desired item is in the top of the recommendation list.
For evaluating popularity bias, we consider the following metrics as used in \cite{abdollahpouri2019managing}:
\begin{figure*}[t]
\subfigure[DN]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.22\textwidth]{plot-target-pop-digi-sea.pdf}}
\subfigure[RR]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-retail-sea.pdf}}
\subfigure[YC-1/64]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-yc1_64-sea.pdf}}
\subfigure[YC-1/4]{\includegraphics[trim={0.250cm 0cm 0.4cm 0cm},clip,width=0.21\textwidth]{plot-target-pop-yc_4-sea.pdf}}
\vspace{-2mm}
\caption{Offline setting evaluation: Recall@20 and MRR@20 with varying $\phi^*$ indicating larger gains by using NISER+ over GNN+ for less popular items. \label{fig:recall-vs-phi}}
\end{figure*}
\begin{figure*}[t]
\subfigure[DN]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.29\textwidth]{plot-day-r-mrr-dn.pdf}}
\subfigure[RR]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.265\textwidth]{plot-day-r-mrr-rr.pdf}}
\subfigure[YC]{\includegraphics[trim={0.0cm 0cm 0.0cm 0cm},clip,width=0.27\textwidth]{plot-day-r-mrr-yc.pdf}}
\vspace{-2mm}
\caption{Online setting evaluation: Recall@20 and MRR@20 for sessions where target item is one of the less popular newly introduced items from the previous day. $f$ denotes the fraction of such sessions in the training set for $\phi^*=0.01$. Standard deviations over five models are shown in lighter-shaded region around the solid lines. \label{fig:plot-online}}
\end{figure*}
\textbf{Average Recommendation Popularity (ARP)}:
This measure calculates the average popularity of the recommended items in each list given by:
\begin{equation}\label{eq:arp}
ARP=\frac{1}{|S|}\sum_{s \in S} \frac{\sum_{i \in L_s}\phi(i)}{K},
\end{equation}
where $\phi(i)$ is popularity of item $i$, i.e. the number of times item $i$ appears in the training set, $L_s$ is the recommended list of items for session $s$, and $|S|$ is the number of sessions in the test set.
An item $i$ belongs to the set $\Gamma_{\phi^*}$ of long-tail items or less popular items if $\frac{\phi(i)}{max_i\phi(i)} \leq \phi^*$.
We evaluate the performance in terms of Recall@20 and MRR@20 for the sessions with target item in the set $\Gamma_{\phi^*}$ by varying $\phi^*$.
\subsection{Results and Observations}
\begin{table}[t]
\caption{Offline setting evaluation: NISER+ versus GNN+ in terms of Average Recommendation Popularity (ARP). Lower values of ARP indicate lower popularity bias. \label{tab:pop-bias}}
\scalebox{0.82}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
GNN+ &495.25$\pm$2.52&453.39$\pm$8.97&4128.54$\pm$27.80&17898.10$\pm$126.93\\
NISER+ &\textbf{487.31$\pm$0.30}&\textbf{398.53$\pm$3.09}&\textbf{3972.40$\pm$41.04}&\textbf{16683.52$\pm$120.74}\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\begin{table}[t]
{\caption{NISER+ versus other benchmark methods in offline setting. Numbers after $\pm$ are standard deviation values over five models. \label{tab:gnn_all_r}}}
\scalebox{0.85}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
\multicolumn{5}{|c|}{\textbf{Recall@20}}\\
\hline
SKNN \cite{jannach2017recurrent}&48.06&56.42&63.77&62.13\\
STAN \cite{garg2019sequence} &50.97&59.80&69.45&70.07\\
\hline
GRU4REC \cite{hidasi2015session}&29.45&-&60.64&59.53\\
NARM \cite{li2017neural}&49.70&-&68.32&69.73\\
STAMP \cite{liu2018stamp}&45.64&53.94&68.74&70.44\\
GNN \cite{wu2018session}&51.39$\pm$0.38&57.63$\pm$0.15&70.54$\pm$0.14&70.95$\pm$0.04\\
GNN+&51.81$\pm$0.11&58.59$\pm$0.10&70.85$\pm$0.08&71.10$\pm$0.07\\
\hline
NIR&52.40$\pm$0.06&60.67$\pm$0.08&71.12$\pm$0.05&71.32$\pm$0.11\\
NISER&52.63$\pm$0.09&60.85$\pm$0.06&70.86$\pm$0.15&71.69$\pm$0.03\\
NISER+&\textbf{53.39$\pm$0.06}&\textbf{61.41$\pm$0.09}&\textbf{71.27$\pm$0.05}&\textbf{71.80$\pm$0.09}\\
\hline\hline
\multicolumn{5}{|c|}{\textbf{MRR@20}}\\
\hline
SKNN \cite{jannach2017recurrent}&16.95&33.16&25.22&24.82\\
STAN \cite{garg2019sequence}&18.48&35.32&28.74&28.89\\
\hline
GRU4REC \cite{hidasi2015session}&8.33&-&22.89&22.60\\
NARM \cite{li2017neural}&16.17&-&28.63&29.23\\
STAMP \cite{liu2018stamp}&14.32&28.49&29.67&30.00\\
GNN \cite{wu2018session}&17.79$\pm$0.16&32.74$\pm$0.09&30.80$\pm$0.09&31.37$\pm$0.13\\
GNN+&18.03$\pm$0.05&33.29$\pm$0.03&30.84$\pm$0.10&31.51$\pm$0.05\\
\hline
NIR &18.52$\pm$0.06&35.57$\pm$0.05&30.99$\pm$0.10&31.73$\pm$0.11\\
NISER &18.27$\pm$0.10&36.09$\pm$0.03&31.50$\pm$0.11&\textbf{31.80$\pm$0.12}\\
NISER+&\textbf{18.72$\pm$0.06}&\textbf{36.50$\pm$0.05}&\textbf{31.61$\pm$0.02}&31.77$\pm$0.10\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\begin{table}
\caption{Ablation results for NISER+ indicating that normalization of embeddings (L$_2$ norm) contributes the most to performance improvement. Here PE: Position Embeddings. \label{tab:gnn_plus_ablation_recall}}
\scalebox{0.85}{
\begin{tabular}{|p{0.2\linewidth}|c|c|c|c|}
\hline
\textbf{Method}&\textbf{DN}&\textbf{RR}&\textbf{YC-1/64}&\textbf{YC-1/4}\\
\hline
\multicolumn{5}{|c|}{\textbf{Recall@20}}\\
\hline
NISER+&\textbf{53.39$\pm$0.06}&\textbf{61.41$\pm$0.09}&\textbf{71.27$\pm$0.05}&71.80$\pm$0.09\\
-L$_2$ norm &52.23$\pm$0.10&59.16$\pm$0.10&71.10$\pm$0.09&71.46$\pm$0.19\\
-Dropout&52.81$\pm$0.12&60.99$\pm$0.09&71.07$\pm$0.13&\textbf{71.90$\pm$0.03}\\
-PE&53.11$\pm0.12$&61.22$\pm$0.03&71.13$\pm$0.04&71.70$\pm$0.11\\
\hline\hline
\multicolumn{5}{|c|}{\textbf{MRR@20}}\\
\hline
NISER+&\textbf{18.72$\pm$0.06}&\textbf{36.50$\pm$0.05}&31.61$\pm$0.02&31.77$\pm$0.10\\
-L$_2$ norm&18.11$\pm$0.05&33.78$\pm$0.04&30.90$\pm$0.07&31.49$\pm$0.07\\
-Dropout&18.43$\pm$0.11&35.99$\pm$0.02&31.56$\pm$0.06&\textbf{31.93$\pm$0.17}\\
-PE &18.60$\pm$0.09&36.32$\pm$0.03&\textbf{31.68$\pm$0.05}&31.71$\pm$0.06\\
\hline
\end{tabular}}
\vspace{-2mm}
\end{table}
\textbf{(1)} \textbf{NISER+ reduces popularity bias in GNN+ in :}
Table \ref{tab:pop-bias} shows that ARP for NISER+ is significantly lower than GNN+ indicating that NISER+ is able to recommend less popular items more often than GNN+, thus reducing popularity bias.
Furthermore, from Fig. \ref{fig:recall-vs-phi}, we observe that NISER+ outperforms GNN+ for sessions with less popular items as targets (i.e. when $\phi^*$ is small), with gains 13\%, 8\%, 5\%, and 2\% for DN, RR, YC-1/64, and YC-1/4 respectively for $\phi^*=0.01$ in terms of Recall@20.
Similarly, gains are 28\%, 18\%, 6\%, and 2\% in terms of MRR@20.
Gains for DN and RR are high as compared to YC. This is due to the high value of $max_i\phi(i)$. If instead we consider $\phi^*=0.001$, gains are as high as 26\% and 9\% for YC-1/64 and YC-1/4 respectively in terms of Recall@20. Similarly, gains are as high as 34\% and 19\% in terms of MRR@20.
We also note that NISER+ is at least as good as GNN+ even for the sessions with more popular items as targets (i.e. when $\phi^*$ is large).
\noindent \textbf{(2)} \textbf{NISER+ improves upon GNN+ in online setting}
for newly introduced items in the set of long-tail items $\Gamma_{\phi^*}$. These items have small number of sessions available for training at the end of the day they are launched.
From Fig. \ref{fig:plot-online}, we observe that for the less popular newly introduced items, NISER+ outperforms GNN+ for sessions where these items are target items on the subsequent day.
This proves the ability of NISER+ to recommend new items on very next day, due to its ability to reduce popularity bias.
Furthermore, for DN and RR, we observe that during initial days, when training data is less, GNN+ performs poorly while performance of NISER+ is relatively consistent across days indicating potential regularization effect of NISER on GNN models in less data scenarios.
Also, as days pass by and more data is available for training, performance of GNN+ improves with time but still stays significantly below NISER+.
For YC, as days pass by, the popularity bias becomes more and more severe (as depicted by very small value for $f$, i.e. the fraction of sessions with less popular newly introduced items) such that the performance of both GNN+ and NISER+ degrades with time. However, importantly, NISER+ still performs consistently better than GNN+ on any given day as it better handles the increasing popularity bias.
\noindent \textbf{(3)} \textbf{NISER and NISER+ outperform GNN and GNN+ in offline setting:}
From Table \ref{tab:gnn_all_r}, we observe that NISER+ shows consistent and significant improvement over GNN in terms of Recall and MRR, establishing a \textit{new state-of-the-art} in SR.
We also conduct an \textbf{ablation study} (removing one feature of NISER+ at a time) to understand the effect of each of the following features of NISER+: i. L$_2$ norm of embeddings, ii. including position embeddings, and iii. applying dropout on item embeddings. As shown in Table \ref{tab:gnn_plus_ablation_recall}, we observe that L$_2$ norm is the most important factor across datasets while dropout and position embeddings contribute in varying degrees to the overall performance of NISER+.
\section{Discussion}
In this work, we highlighted that the typical item-frequency distribution with long tail leads to popularity bias in state-of-the-art deep learning models such as GNNs \cite{wu2018session} for session-based recommendation.
We then argued that this is partially related to the `radial' property of the softmax loss that, in our setting, implies that the norm for popular items will likely be larger than the norm of less popular items.
We showed that learning the representations for items and session-graphs by optimizing for cosine similarity instead of inner product can help mitigate this issue to a large extent.
Importantly, this ability to reduce popularity bias is found to be useful in the online setting where the newly introduced items tend to be less popular and are poorly modeled by existing approaches.
We observed significant improvements in overall recommendation performance by normalizing the item and session-graph representations and improve upon the existing state-of-the-art results.
In future, it would be worth exploring NISER to improve other algorithms like STAMP \cite{liu2018stamp} that rely on similarity between embeddings for items, sessions, users, etc.
\bibliographystyle{ACM-Reference-Format}
| train/arxiv |
BkiUaXnxK5YsWTJsNBiq | 5 | 1 | \section{Introduction}
The classical theorem of B\'ezout~\cite{Bezout} bounds the number of solutions
to a system of polynomials by the product of their degrees.
While this B\'ezout bound is sharp for generic systems of polynomial equations,
that is no longer the case when the equations possess additional structure.
For example, Kushnirenko~\cite{BKK} showed that if all the polynomials have
the same Newton polytope, then the number of nondegenerate solutions to
such a system is at most the volume of the Newton polytope, suitably normalized.
Bounds for the number of nondegenerate real solutions are governed by
Kushnirenko's ``fewnomial principle'': roughly, few monomials implies few
solutions or restricted topology~\cite{Kush_note}.
This principle was established by Khovanskii in his fundamental work on
fewnomials~\cite{Kh91} in which he showed that a system of $n$ polynomials in $n$ variables
where the polynomials have $1{+}k{+}n$ distinct monomials has fewer than
\begin{equation}\label{Eq:Khovanskii_bound}
2^{\binom{k{+}n}{2}}\, (n{+}1)^{k{+}n}
\end{equation}
nondegenerate positive solutions.
This bound is remarkable as it is independent of the degrees and Newton polytopes of the
polynomials, which control the number of complex solutions.
Few of the complex solutions to a fewnomial system can be real.
Khovanskii's bound was lowered by Bihan and Sottile~\cite{BS_Khovanskii} to
\begin{equation}\label{Eq:BS}
\frac{e^2{+}3}{4}\, 2^{\binom{k}{2}}\, n^k\,.
\end{equation}
For this, they transformed the original polynomial system into an
equivalent Gale dual~\cite{BS_Gale} system of rational functions, whose number of solutions
they bounded.
An essential step for the bound on Gale systems uses Khovanskii's generalization of the
classical Rolle Theorem.
The bound~\eqref{Eq:BS} is smaller than Khovanskii's
bound~\eqref{Eq:Khovanskii_bound} because Khovanskii's bound is a specialization to
polynomials of a bound for more general functions and the proof
of~\eqref{Eq:BS} takes advantage of some special geometry enjoyed by polynomials.
We derive bounds of fewnomial type for polynomial systems with structure
intermediate between that of fewnomials and general polynomials.
These bounds can be dramatically smaller than the fewnomial bound~\eqref{Eq:BS}, and like
that bound do not depend upon the degrees of the polynomials involved.
A collection ${\mathcal{A}}\subset\mathbb{Z}^n$ of exponent vectors is
\demph{$(d,\ell)$-dense} if there are integers $d,\ell$ such that
${\mathcal{A}}$ admits a decomposition of the form
\begin{equation}\label{Eq:strFew}
{\mathcal{A}}\ =\ \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\; \bigcup\; {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}$ consists of $n$ affinely independent vectors, $\psi\colon\mathbb{Z}^\ell\to\mathbb{Z}^n$
is an affine-linear map, and $\includegraphics{figures/Simplex.eps}^\ell$ is the unit simplex in $\mathbb{R}^\ell$.
A Laurent polynomial whose support ${\mathcal{A}}$ is $(d,\ell)$-dense~\eqref{Eq:strFew} is a
\demph{$(d,\ell)$-dense fewnomial}.
We show that a system of $(d,\ell)$-dense fewnomials
in $n$ variables has fewer than
\begin{equation}\label{Eq:structured_bound}
\frac{e^2{+}3}{4}\, 2^{\binom{\ell }{2}}\, n^\ell \cdot d^\ell
\end{equation}
nondegenerate positive solutions.
To compare this to~\eqref{Eq:BS}, observe that $k{+}1\ =\ \binom{d+\ell}{\ell}$,
so the bounds coincide when $d=1$, but otherwise~\eqref{Eq:structured_bound} is dramatically
smaller.
In~\cite{BBS_all}, the methods of~\cite{BS_Khovanskii} were extended to establish the bound
\begin{equation}\label{Eq:BBS_bound}
\frac{e^{\Red{4}}+3}{4}\, 2^{\binom{k}{2}}\, n^k\,,
\end{equation}
for the number of nonzero real solutions to a fewnomial system---not just positive
solutions---when the exponent vectors span a sublattice of $\mathbb{Z}^n$ of odd index.
The same arguments show that if the exponent vectors ${\mathcal{A}}$ span a
sublattice of $\mathbb{Z}^n$ of odd index, then the number of nondegenerate real solutions to a system
of $(d,\ell)$-dense fewnomials is at most
\begin{equation}\label{Eq:structured_real}
\frac{e^{\Red{4}}+3}{4}\, 2^{\binom{\ell }{2}}\, n^\ell \cdot d^\ell\,.
\end{equation}
Khovanskii also gave a bound for the sum \demph{$b_*(X)$} of the Betti
numbers of a smooth hypersurface in the positive orthant $\mathbb{R}^n_>$ defined by a polynomial
with $1{+}k{+}n$ monomial terms~\cite{Kh91} (Corollary 4, p. 91),
\begin{equation}\label{eq:Khovanskii}
b_*(X)\ \leq\ (2n^2{-}n{+}1)^{k{+}n}\, (2n)^{n{-}1}\, 2^{\binom{k{+}n}{2}}\,.
\end{equation}
Bihan and Sottile~\cite{BS_Betti} used the fewnomial bound and stratified Morse theory for
a manifold with corners~\cite{GM88} to lower this to
\begin{equation}\label{eq:Betti}
b_*(X) \ <\ \frac{e^2{+}3}{4}\, 2^{\binom{k}{2}}
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^k\,.
\end{equation}
The same arguments show that when $X$ is defined by a $(d,\ell)$-dense fewnomial, we have
\begin{equation}\label{Eq:structured_Betti}
b_*(X)\ <\ \frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\end{equation}
An important step in these arguments is a version of Gale duality for
dense fewnomials, which generalizes Gale duality for polynomial
systems as established in~\cite{BS_Gale}.
These bounds~\eqref{Eq:structured_bound},~\eqref{Eq:structured_real},
and~\eqref{Eq:structured_Betti} simultaneously generalize the results
of~\cite{BS_Khovanskii},~\cite{BBS_all},
and~\cite{BS_Betti}, which are the cases when $d=1$.
The case of $\ell=1$ of the bound~\eqref{Eq:BBS_bound} was established
in~\cite{BBS_few}, where a $(d,1)$-dense fewnomial was called a near circuit.
This paper is structured as follows.
We begin in Section~\ref{Sec:one} with definitions and examples of dense fewnomials, give the
precise statements of our main theorems, and study an example when $n=2$.
Section~\ref{S:gale} is devoted to establishing the variant of Gale duality appropriate for
dense fewnomials, and in Section~\ref{S:bounds}, which is the heart of this paper, we establish
the bounds~\eqref{Eq:structured_bound} and~\eqref{Eq:structured_real}.
We develop the necessary tools and give the proof of our bound~\eqref{Eq:structured_Betti}
for the sum of Betti numbers in Section~\ref{Sec:Betti}.
We thank Maurice Rojas, who suggested looking for extentions of the fewnomial bound as a
class project to Rusek and Shakalli.
\section{Dense Fewnomials}\label{Sec:one}
An integer vector $\alpha=(a_1,\dotsc,a_n)\in\mathbb{Z}^n$ is the exponent of a Laurent
monomial,
\[
x^\alpha\ :=\ x_1^{a_1} x_2^{a_2}\dotsb x_n^{a_n}\,.
\]
A polynomial $f$ with \demph{support} ${\mathcal{A}}\subset\mathbb{Z}^n$ is one whose exponent vectors
lie in ${\mathcal{A}}$,
\[
f\ =\ \sum_{\alpha\in{\mathcal{A}}} a_\alpha x^\alpha
\qquad(a_\alpha\in\mathbb{R})\,.
\]
We are interested in systems of $n$ polynomials, each with support ${\mathcal{A}}$.
We obtain novel bounds on the number of real solutions to a system of polynomials when the set
of exponent vectors has structure that is intermediate between fewnomials
and dense polynomials.
Kushnirenko~\cite{BKK} showed that a general system of polynomials, all with support ${\mathcal{A}}$,
will have $n!\operatorname{vol}(\operatorname{conv}({\mathcal{A}}))$ complex solutions, the normalized volume of the convex hull of
the exponent vectors.
While this is also a bound for the number of real solutions, there is another bound which
depends only upon the number of exponents.
Specifically, a \demph{fewnomial system} is one in which the support ${\mathcal{A}}$ consists of
$1{+}k{+}n$ monomials, but is otherwise unstructured, and such a system has the
bound~\eqref{Eq:BS} on its number of positive solutions.
When the exponents ${\mathcal{A}}$ affinely span a sublattice of odd index in $\mathbb{Z}^n$, a fewnomial
system has the bound~\eqref{Eq:BBS_bound} for its number of nondegenerate nonzero real solutions.
A \demph{dense fewnomial} is a polynomial whose support ${\mathcal{A}}$ is intermediate
between fewnomials and general polynomials in the following way.
Let $d,\ell,n$ be positive integers, $\includegraphics{figures/Simplex.eps}^\ell\subset\mathbb{R}^\ell$ be the standard
unit simplex, the convex hull of the original and the $\ell$ unit basis vectors, and
$\psi\colon\mathbb{Z}^\ell\to\mathbb{Z}^n$ be an affine-linear map.
A \demph{$(d,\ell)$-dense fewnomial} is a Laurent polynomial $f$ whose support
${\mathcal{A}}\subset\mathbb{Z}^n$ admits a decomposition
\begin{equation}\label{Eq:near_polyhedral}
{\mathcal{A}}\ =\ \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\;\bigcup\; {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}=\{w_1,\dotsc,w_n\}$ consists of $n$ affinely independent vectors.
Such a set ${\mathcal{A}}$~\eqref{Eq:near_polyhedral} is \demph{$(d,\ell)$-dense}.
We give some examples of $(d,\ell)$-dense sets of exponent vectors.
\begin{enumerate}
\item Any collection ${\mathcal{A}}$ of $1{+}\ell{+}n$ exponent vectors that affinely spans
$\mathbb{R}^n$ is $(1,\ell)$-dense.
To see this, let ${\mathcal{W}}\subset{\mathcal{A}}$ be $n$ affinely independent vectors.
Writing ${\mathcal{A}}-{\mathcal{W}}=\{v_0,v_1,\dotsc,v_\ell\}$, these vectors
are the image of the integer points in $\includegraphics{figures/Simplex.eps}^\ell$ under the affine map $\psi$ that
takes the $i$th unit vector to $v_i$ and the origin to $v_0$.
Thus ordinary fewnomials with $1{+}k{+}n$ monomials are $(1,k)$-dense fewnomials.
\item When $\ell=1$, the exponent vectors ${\mathcal{A}}$ of a dense
fewnomial form a \demph{near circuit} in the terminology of~\cite{BBS_few}.
There, it was shown that if ${\mathcal{A}}$ spans $\mathbb{Z}^n$, then a system with support ${\mathcal{A}}$
has at most $2dn{+}1$ nonzero real solutions.
\item A general $(d,\ell)$-dense set ${\mathcal{A}}$ has the following form,
\begin{equation}\label{Eq:general_form}
{\mathcal{A}}\ :=\
\Bigl\{ v_0\ +\ \sum_{m=1}^\ell \lambda_mv_m \mid
0\leq \lambda_m\,,\ \sum_i\lambda_i\leq d \Bigr\}
\bigcup {\mathcal{W}}\,,
\end{equation}
where ${\mathcal{W}}=\{w_1,\dotsc,w_n\}\subset\mathbb{Z}^n$ is affinely independent and
$v_0,v_1,\dotsc,v_\ell$ are integer vectors.
Below is an example of such a set ${\mathcal{A}}$ in $\mathbb{Z}^2$ ($v_0=(0,0)$ is the open circle).
\begin{equation}\label{Eq:triangle}
\raisebox{-38pt}{\begin{picture}(157,76)(-11,0)
\put(0,0){\includegraphics{figures/triangle.eps}}
\put(-11,1){$v_0$} \put(64,21){$v_1$}
\put(7,32){$v_2$}
\put(18,63){$w_2$} \put(98,1){$w_1$}
\end{picture}}
\end{equation}
For this, $n=\ell=d=2$, ${\mathcal{W}}=\{(9,0),(2,7)\}$ and
$v_1=(7,1)$ and $v_2=(2,3)$.
Here is a $(2,3)$-dense set in $\mathbb{Z}^2$.
\[
\begin{picture}(187,106)(-11,0)
\put(0,0){\includegraphics{figures/tetrahedron.eps}}
\put(-3,32){$v_2$} \put(78,13){$v_1$}
\put(45,53){$v_3$} \put(-11,1){$v_0$}
\put(8,89){$w_2$} \put(138,13){$w_1$}
\end{picture}
\]
\end{enumerate}
\begin{theorem}\label{Th:Structured_bound}
Suppose that ${\mathcal{A}}\subset\mathbb{Z}^n$ is $(d,\ell)$-dense.
Then a system
\[
f_1(x_1,\dotsc,x_n)\ =\
f_2(x_1,\dotsc,x_n)\ =\ \dotsb \ =\
f_n(x_1,\dotsc,x_n)\ =\ 0
\]
of real polynomials with support ${\mathcal{A}}$ has fewer than
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell
\]
nondegenerate positive solutions.
If the affine span of ${\mathcal{A}}$ is a sublattice of\/ $\mathbb{Z}^n$ with odd index, then
the number of nondegenerate real solutions is less than
\[
\frac{e^4{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
\end{theorem}
The bounds in Theorem~\ref{Th:Structured_bound} hold if the support of the system is only a
subset of ${\mathcal{A}}$, for in that case, we may perturb the coefficients of the system to obtain
one whose support is exactly ${\mathcal{A}}$ without decreasing the number of nondegenerate solutions.
When $n=d=\ell=2$, this bound for positive solutions is $83$.
This is already significant for the $(2,2)$-dense set~\eqref{Eq:triangle}.
Since the Kushnirenko bound for~\eqref{Eq:triangle} is 112,
a general system with support~\eqref{Eq:triangle} will have 112 complex solutions, at most 83
of which can be positive.
A generic system of $(d,\ell)$-dense fewnomials (as in~\eqref{Eq:near_polyhedral}),
\begin{equation}\label{Eq:structured_system}
f_1(x)\ =\ f_2(x)\ =\ \dotsb\ =\ f_n(x)\ =\ 0\,,
\end{equation}
will have an invertible matrix of coefficients of the monomials
$\{x^{w_i}\mid i=1,\dotsc,n\}$, and so we may solve~\eqref{Eq:structured_system} for these
monomials to get the equivalent system
\begin{equation}\label{Eq:diagonal}
x^{w_i}\ =\ \sum_{p\in d\includegraphics{figures/Simplex.sm.eps}^\ell\cap\mathbb{Z}^\ell} a_{i,p} x^{\psi(p)}\,,
\qquad\mbox{for}\quad i=1,\dotsc,n,.
\end{equation}
For each $i=1,\dotsc,n$, define the degree $d$ polynomial in variables $y\in\mathbb{R}^\ell$,
\[
\dcol{h_i(y)}\ :=\ \sum_{p\in d\includegraphics{figures/Simplex.sm.eps}^\ell\cap\mathbb{Z}^\ell} a_{i,p} y^{p}\,.
\]
Following the notation in~\eqref{Eq:general_form}, we translate the set ${\mathcal{A}}$ by $-v_0$,
which amounts to multiplying the equations~\eqref{Eq:structured_system}
and~\eqref{Eq:diagonal} by $x^{-v_0}$, and does not change their solutions.
Thus we may assume that $v_0=0$ so that $\psi$ is a linear map, and then let
${\mathcal{V}}:=\{v_1,\dotsc,v_\ell\}\subset\mathbb{Z}^n$ be the images of the standard basis vectors
under $\psi$.
Then we have $x^{w_i}=h_i(x^{v_1},\dotsc,x^{v_\ell})$.
A linear relation among
the vectors in ${\mathcal{V}}$ and ${\mathcal{W}}$,
\begin{equation}\label{Eq:lin_reln}
\sum_{m=1}^\ell b_m v_m\ +\ \sum_{i=1}^n c_i w_i\ =\ 0\,,
\end{equation}
implies the multiplicative relation among the monomials
\[
\prod_{m=1}^\ell (x^{v_m})^{b_m}\ \cdot\ \prod_{i=1}^n (x^{w_i})^{c_i}\ =\ 1\,.
\]
If we use~\eqref{Eq:diagonal} to first substitute $h_i(x^{v_1},\dotsc,x^{v_\ell})$
for $x^{w_i}$ for $i=1,\dotsc,n$ in this expression,
and then substitute $y_m$ for $x^{v_m}$, for $m=1,\dotsc,\ell$, we obtain
\[
\prod_{m=1}^\ell(y_m)^{b_m}\ \cdot\ \prod_{i=1}^n (h_i(y))^{c_i}\ =\ 1\,.
\]
Write $\beta=(b_1,\dotsc,b_\ell)$ for the vector of the coefficients of ${\mathcal{V}}$
in~\eqref{Eq:lin_reln} and $\gamma=(c_1,\dotsc,c_n)$ for the vector of the coefficients of
${\mathcal{W}}$.
Then we may write the left hand side of this last expression compactly as
$y^\beta\cdot h(y)^\gamma$.
Now suppose that $(\beta_j,\gamma_j)\in\mathbb{Z}^\ell\oplus\mathbb{Z}^n$ for $j=1,\dotsc,\ell$ is a basis for
the $\mathbb{Z}$-module of linear relations among the vectors in ${\mathcal{V}}\cup{\mathcal{W}}$.
Then the system
\begin{equation}\label{Eq:str_Gale}
y^{\beta_j}\ \cdot\ h(y)^{\gamma_j}
\ =\ 1\,,\qquad j=1,\dotsc,\ell
\end{equation}
is a \demph{$(d,\ell)$-dense Gale system} dual to the original
system~\eqref{Eq:structured_system} of polynomials.
\begin{theorem}\label{Th:GaleDual}
Let~$\eqref{Eq:structured_system}$ be a system of $(d,\ell)$-dense fewnomials
and~$\eqref{Eq:str_Gale}$ be its corresponding dual $(d,\ell)$-dense Gale system.
Then the number of nondegenerate positive solutions to~$\eqref{Eq:structured_system}$ is equal
to the number of nondegenerate positive solutions to~$\eqref{Eq:str_Gale}$ where
$h_i(y)>0$ for each $i=1,\dotsc,n$.
If the exponents ${\mathcal{A}}$ affinely span a sublattice of\/ $\mathbb{Z}^n$ of odd index and the relations
$(\beta^j,\gamma^j)\in\mathbb{Z}^\ell\oplus\mathbb{Z}^n$ for $j=1,\dotsc,\ell$ span a sub $\mathbb{Z}$-module of odd index
in the module of all linear relations, then the number of nondegenerate real solutions
to~$\eqref{Eq:structured_system}$ is equal to the number of nondegenerate real solutions
to~$\eqref{Eq:str_Gale}$ in $(\mathbb{R}^\times)^\ell$ where no $h_i(y)$ vanishes.
\end{theorem}
This follows from Theorem~\ref{Th:Str_Gale} on Gale duality for $(d,\ell)$-dense fewnomials.
Thus we may prove Theorem~\ref{Th:Structured_bound} by establishing bounds for
$(d,\ell)$-dense Gale systems.\smallskip
Let us consider an example of this duality.
The system of Laurent polynomials,
\begin{equation}\label{Eq:Laurent_system}
\begin{array}{rcl}
\Color{1 0 1 0.4}{f}\ :=\
27t^{-5}\ +\ 31 - 16t^2u - 16t^2u^{-1} -16t^4u^2+ 40t^4-16t^4u^{-2}&=& 0\,,\\
\Color{0 1 1 0.3}{g}\ :=\
12t\hspace{15.5pt}+\ 40 -32t^2u -32t^2u^{-1}
+\ \,5t^4u^2+\ \,6t^4+\ \,5t^4u^{-2}&=& 0\,,\rule{0pt}{14pt}
\end{array}
\end{equation}
has 36 complex solutions with nonzero coordinates,
ten of which are real and eight of which lie in the positive quadrant.
We show the curves $\Color{1 0 1 0.4}{f=0}$ and $\Color{0 1 1 0.3}{g=0}$ defined by the
polynomials~\eqref{Eq:Laurent_system}.
In the picture on the left, the horizontal scale has been exaggerated.
Its shaded region is shown on the right, where now the vertical scale is exaggerated.
\begin{equation}\label{Eq:Laurent_Curves}
%
\raisebox{-60pt}{
\begin{picture}(240,120)
\put(0,0){\includegraphics[height=120pt]{figures/poly_big.eps}}
\put(236,3){$t$} \put(121,110){$u$}
\put(68,88){$\Color{0 1 1 0.3}{g}$} \put(88,88){$\Color{0 1 1 0.3}{g}$}
\put(138,88){$\Color{0 1 1 0.3}{g}$} \put(180,44){$\Color{0 1 1 0.3}{g}$}
\put(1,49){$\Color{1 0 1 0.4}{f}$} \put(4,44){\Color{1 0 1 0.4}{\vector(0,-1){25}}}
\put(228,14){$\Color{1 0 1 0.4}{f}$}
\end{picture}
\qquad
\begin{picture}(141,120)
\put(133,6){$t$} \put(13,110){$u$}
\put(0,0){\includegraphics[height=120pt]{figures/poly_sm.eps}}
\put(60,31){$\Color{0 1 1 0.3}{g}$} \put(117,40){$\Color{0 1 1 0.3}{g}$}
\put(133,104){$\Color{1 0 1 0.4}{f}$}
\end{picture}}
\end{equation}
Here are numerical approximations to the ten real solutions.
\begin{eqnarray*}
&( 0.619, 0.093),\,
( 0.839, 0.326),\,
( 1.003, 0.543),\,
( 1.591, 0.911),\,
(-1.911, 0.864),\,&\\
&( 0.619, 10.71),\,
( 0.839, 3.101),\,
( 1.003, 1.843),\,
( 1.591, 1.097),\,
(-1.911, 1.158)\,.&
\end{eqnarray*}
(The repetition in the $t$-coordinates is explained by the symmetry $u\mapsto u^{-1}$
of~\eqref{Eq:Laurent_system}.)
The system~\eqref{Eq:Laurent_system} is a system of $(2,2)$-dense fewnomials, as we may see
from its support.
\[
\begin{picture}(206,108)
\put(0,0){\includegraphics{figures/Laurent_supp.eps}}
\put(197,65){$t$} \put(89,95){$u$}
\put(129 ,76){$v_2$} \put(131,28){$v_1$}
\put(124.5,46){$w_2$} \put( 14,45){$w_1$}
\end{picture}
\]
If we solve~\eqref{Eq:Laurent_system} for the monomials $t^{-5}$ and $t$, we obtain
\begin{equation}\label{Eq:solved_system}
\begin{array}{rcl}
t^{-5} &=&\frac{1}{27}(-\ 31 + 16t^2u + 16t^2u^{-1}+16t^4u^2- 40t^4+16t^4u^{-2})
\ =:\ h_1(t^2u,t^2u^{-1})\\
t\hspace{11pt}&=&\frac{1}{12}(-\ 40 +32t^2u + 32t^2u^{-1}
-\ \,5t^4u^2-\ \,6t^4-\ \hspace{1.5pt}5t^4u^{-2})
\ =:\ h_2(t^2u,t^2u^{-1}) \,.\rule{0pt}{14pt}
\end{array}
\end{equation}
We convert this into a $(d,\ell)$-dense Gale system dual to~\eqref{Eq:Laurent_system}.
First observe that
\begin{equation}\label{Eq:monoms}
\begin{array}{rcl}
\left(t^2u\right)^{\Red{1}} \left(t^2u^{-1}\right)^{\Red{1}}
\left(t^{-5}\right)^{\Red{1}} \left(t\right)^{\Red{1}} \hspace{6.5pt}
&=&1\\
\left(t^2u\right)^{\Red{2}} \left(t^2u^{-1}\right)^{\Red{2}}
\left(t^{-5}\right)^{\Red{1}} \left(t\right)^{\Red{-3}}
&=&1\,.\rule{0pt}{14pt}
\end{array}
\end{equation}
We use the equations~\eqref{Eq:solved_system} to replace the monomials
$t^{-5}$ and $t$ in~\eqref{Eq:monoms} and then apply the substitutions
$x:=t^2u$ and $y:=t^2u^{-1}$ (so that $x^2=t^4u^2$, $xy=t^4$, and $y^2=t^4u^{-2}$).
Then, after clearing denominators and
rearranging, we have
\begin{equation}\label{Eq:Gale}
\begin{array}{rcl}
x^{\Red{1}} y^{\Red{1}} h_1(x,y)^{\Red{1}} h_2(x,y)^{\Red{1}}
\ -\ 1
&=&0 \\
x^{\Red{2}} y^{\Red{2}} h_1(x,y)^{\Red{1}}
\ -\ h_2(x,y)^{\Red{3}}
&=&0 \,.\rule{0pt}{15pt}
\end{array}
\end{equation}
This system has 36 complex solutions, ten of which are real and eight of which lie in the
shaded region in the picture below where $x,y,h_1(x,y),$ and $h_2(x,y)$ are all positive.
\begin{equation}\label{Eq:Gale_Curves}
\raisebox{-90pt}{
\begin{picture}(275,190)(-70,0)
\put(0,0){\includegraphics{figures/Gale.eps}}
\put(-62.5,65){$h_2(x,y)=0$}
\put(0,68){\vector(1,0){45}}
\put(-62.5,47.6){$h_1(x,y)=0$} \put(141,150){$h_1(x,y)=0$}
\put(155,145){\vector(-1,-1){44}}
\put(0,50.5){\vector(4,-1){67}}
\put(71,3){$x=0$} \put(-30,26){$y=0$}
\put(69,6){\vector(-1,0){29}}
\end{picture}
}
\end{equation}
The numbers of complex, real, and suitably positive solutions to the
two systems~\eqref{Eq:Laurent_system} and~\eqref{Eq:Gale}
is a consequence of Theorem~\ref{Th:Str_Gale} on
structured Gale duality.
Here are numerical approximations to the ten real solutions of~\eqref{Eq:Gale}
\begin{eqnarray*}
&(4.229, 3.154),\,
(4.098, 0.036),\,
(2.777, 2.306),\,
(2.184, 0.227),\,
(1.853, 0.546),\,&
\\
&(3.154, 4.229),\,
(0.036, 4.098),\,
(2.306, 2.777),\,
(0.227, 2.184),\,
(0.546, 1.853)\,.&
\end{eqnarray*}
We remark that there is no relation between the two pairs of curves
in~\eqref{Eq:Laurent_Curves} and~\eqref{Eq:Gale_Curves}.
Gale duality only
asserts a scheme-theoretic equality between the points of intersection of each pair of curves.
\section{Gale Duality for $(d,\ell)$-dense fewnomials}\label{S:gale}
Gale duality~\cite{BS_Gale} asserts that a system of $n$
polynomials in $n$ variables involving a total of $1{+}k{+}n$ distinct monomials is
equivalent to a system of $k$ rational functions of a particular form in the complement of
an arrangement of $k{+}n$ hyperplanes in $\mathbb{R}^k$.
A modification of Gale duality asserts that a system of $(d,\ell)$-dense fewnomials is
equivalent to a system $\ell\; (\leq k)$ rational functions in the complement ${\mathcal{M}}(\mathbb{R})$ of the
coordinate axes of $\mathbb{R}^\ell$ and of $n$ degree $d$ hypersurfaces.
We will call such a system a $(d,\ell)$-dense Gale system.
Write $\dcol{\mathbb{T}}$ for the non-zero complex numbers, $\mathbb{C}^\times$.
\subsection{$(d,\ell)$-dense polynomials}
\label{S:nearpolyhedral}
Suppose that ${\mathcal{A}}=\psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\cup{\mathcal{W}}$ is $(d,\ell)$-dense and
that it affinely spans $\mathbb{Z}^n$.
Translating ${\mathcal{A}}$ by $\psi(0)$ if necessary, we may assume that $\psi$ is linear.
Write ${\mathcal{V}}=\{v_1,\dotsc,v_\ell\}\subset\mathbb{Z}^n$ for the images under $\psi$ of the standard
basis vectors of $\mathbb{Z}^\ell$ and list the elements of ${\mathcal{W}}$ as $\{w_1,\dotsc,w_n\}$.
Consider the map
\[
\begin{array}{rclcl}
\varphi&\colon&\mathbb{T}^n&\longrightarrow&\mathbb{T}^\ell\times\mathbb{T}^n\\
&&x&\longmapsto&(x^{v_1},\dotsc,x^{v_\ell},\,x^{w_1},\dotsc,x^{w_n})\,.
\rule{0pt}{14pt}
\end{array}
\]
Write $y=(y_1,\dotsc,y_\ell)$ for the coordinates of the first factor $\mathbb{T}^\ell$ and
$z=(z_1,\dotsc,z_n)$ for the coordinates of the second factor $\mathbb{T}^n$.
A polynomial with support ${\mathcal{A}}$ has the form
\begin{eqnarray}
f&=& \sum_{\substack{\lambda=(\lambda_1,\dotsc,\lambda_\ell)\\|\lambda|=d}}
a_\lambda x^{\lambda_1 v_1}\dotsb x^{\lambda_\ell v_\ell}\ +\
\sum_{i=1}^n a_ix^{w_i}\nonumber\\
&=& \varphi^*\Bigl(\sum_{|\lambda|=d} a_\lambda y_1^{\lambda_1}\dotsb y_\ell^{\lambda_\ell}\ +\
\sum_{i=1}^n a_i z_i\Bigr)
\ =\ \varphi^*(h(y) + \Lambda(z))\,,\label{Eq:pullback}
\end{eqnarray}
the pullback along $\varphi$ of a polynomial $h$ of degree $d$ and a linear form $\Lambda$.
\subsection{Proof of Theorem~\ref{Th:GaleDual}}
Let ${\mathcal{A}}$ and ${\mathcal{W}}$ be as in the previous subsection.
Suppose that $f_1(x)=\dotsb=f_n(x)=0$ is a system of $(d,\ell)$-dense fewnomials with
support ${\mathcal{A}}$.
By~\eqref{Eq:pullback}, there exist polynomials $h_1(y),\dotsc,h_n(y)$ of degree $d$ in
the variables $y=(y_1,\dotsc,y_\ell)$ and linear forms $\Lambda_1(z),\dotsc,\Lambda_n(z)$
in variables $z=(z_1,\dotsc,z_n)$ such that
\[
f_i(x)\ =\ \varphi^*( h_i(y) + \Lambda_i(z))
\qquad i=1,\dotsc,n\,.
\]
Since we wish to enumerate non-degenerate solutions, we may assume that the polynomials
$h_i(y),\Lambda_i(z)$ are generic, for perturbing the coefficients of the $f_i$ can only
increase their number of non-degenerate solutions.
Thus we may assume that $\Lambda_1(z),\dotsc,\Lambda_n(z)$ are linearly independent.
Replacing the polynomials $f_1,\dotsc,f_n$ by appropriate linear combinations, we may
assume that $\Lambda_i(z)=-z_i$ for each $i$.
Then our system becomes
\begin{equation}\label{Eq:graph_pullback}
\varphi^*(h_1(y)-z_1)\ =\
\varphi^*(h_2(y)-z_2)\ =\ \dotsb\ =\
\varphi^*(h_n(y)-z_n)\ =\ 0\,.
\end{equation}
If we define $H\subset\mathbb{C}^\ell\times\mathbb{C}^n$ by the equations
\[
H\ =\ \{(y,z)\in\mathbb{C}^\ell\times\mathbb{C}^n\mid
z_1=h_1(y),\dotsc,z_n=h_n(y)\}\,,
\]
then our system~\eqref{Eq:graph_pullback} has the alternative geometric description
as $\varphi^*(H)$.
Since $\mathbb{Z}{\mathcal{A}}=\mathbb{Z}^n$, $\varphi$ is an isomorphism onto its image, and we deduce the
following lemma.
\begin{lemma}
The system~$\eqref{Eq:graph_pullback}$ is isomorphic to the intersection
$\varphi(\mathbb{T}^n)\cap H$ in $\mathbb{C}^\ell\times\mathbb{C}^n$.
\end{lemma}
This is the first step in Gale duality for $(d,\ell)$-dense fewnomials.
For the second step observe that $H$ is isomorphic to $\mathbb{C}^\ell$, as it
is the graph of the function $\mathbb{C}^\ell\to\mathbb{C}^n$ given by $y\mapsto(h_1(y),\dotsc,h_n(y))$.
Let $\Psi\colon\mathbb{C}^\ell\to H$ be the isomorphism between $\mathbb{C}^\ell$ and this graph.
Then the system~\eqref{Eq:graph_pullback} is equivalent to
\[
\varphi(\mathbb{T}^n)\cap \Psi(\mathbb{C}^\ell)\,.
\]
We determine the equations in $\mathbb{T}^\ell\times\mathbb{T}^n$ that define $\varphi(\mathbb{T}^n)$.
For $\beta=(b_1,\dotsc,b_\ell)\in\mathbb{Z}^\ell$ and $\gamma=(c_1,\dotsc,c_n)\in\mathbb{Z}^n$, let
\[
y^\beta\cdot z^\gamma\ =\ y_1^{\beta_1}\dotsb y_\ell^{\beta_\ell}
\cdot z_1^{c_1}\dotsb z_n^{c_n}\,.
\]
We similarly write $h(y)^\gamma$ for $h_1(y)^{c_1}\dotsb h_n(y)^{c_n}$.
Suppose that ${\mathcal{B}}\subset\mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$
is a basis for the $\mathbb{Z}$-linear relations among the exponent vectors ${\mathcal{V}}\cup{\mathcal{W}}$.
As ${\mathcal{A}}$ spans $\mathbb{R}^n$, so does ${\mathcal{V}}\cup{\mathcal{W}}$, and so ${\mathcal{B}}$ consists of
$\ell$ vectors, $\{(\beta_1,\gamma_1),\dotsc,(\beta_\ell,\gamma_\ell)\}$.
Then the image
$\varphi(\mathbb{T}^{n})\subset \mathbb{T}^\ell\times \mathbb{T}^n$ is
the subtorus defined by
\begin{equation}\label{Eq:sGale}
y^{\beta_j}\cdot z^{\gamma_j}\ =\ 1 \qquad\mbox{for } j=1,\dotsc,\ell\,.
\end{equation}
\begin{proposition}\label{prop:npGale}
The pullback of $\varphi(\mathbb{T}^{n})\cap H$ along the map
$\Psi$ is the system
\begin{equation}\label{Eq:npmf}
y^{\beta_j} \cdot h(y)^{\gamma_j}\ =\ 1
\qquad\mbox{for } j=1,\dotsc,\ell\,.
\end{equation}
This is well-defined in $\mathbb{C}^{\ell}$ in the complement ${\mathcal{M}}(\mathbb{C})$ of the
coordinate planes and the hypersurfaces $h_i(y)=0$ for $i=1,\dotsc,n$.
\end{proposition}
We may now state our main theorem on structured Gale duality.
The \demph{saturation} of a submodule ${\mathcal{B}}\subset\mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$ is the set
$({\mathcal{B}}\otimes_\mathbb{Z}\mathbb{R}) \cap \mathbb{Z}^{\ell}\oplus\mathbb{Z}^n$ of integer points in its linear span.
\begin{theorem}\label{Th:Str_Gale}
Suppose that ${\mathcal{A}},{\mathcal{W}},\psi$, and ${\mathcal{V}}$ are as above and that ${\mathcal{A}}$
spans $\mathbb{Z}^{n}$.
Then the solution set to~\eqref{Eq:graph_pullback} in $\mathbb{T}^{n}$
is scheme-theoretically isomorphic to the solution set of the system of
rational functions~\eqref{Eq:npmf} defined in $\mathbb{T}^\ell$ in the
complement of the hypersurfaces $h_i(y)=0$, for $i=1,\dotsc,n$.
If the coefficients of the polynomials $f_i$ are real, then so are those of $h_i(y)$.
If the span of ${\mathcal{A}}$ has odd index in $\mathbb{Z}^{n}$ and the integer span of the
exponents ${\mathcal{B}}$ has odd index in its saturation, then the analytic subscheme
defined in $(\mathbb{R}^\times)^{n}$ by~\eqref{Eq:graph_pullback} is isomorphic to the analytic
subscheme defined by~\eqref{Eq:npmf} in the complement of the
hypersurfaces $h_i(y)=0$ in $(\mathbb{R}^\times)^\ell$.
If now the exponents ${\mathcal{A}}$ only span a full rank sublattice of
$\mathbb{Z}^{n}$ and the exponents ${\mathcal{B}}$ only span a full rank sublattice of
the module of linear relations among ${\mathcal{V}}\cup{\mathcal{W}}$, then the analytic subscheme of\/
$\mathbb{R}_{>0}^{n}$ defined by~\eqref{Eq:graph_pullback} is isomorphic to the analytic subscheme
defined by~\eqref{Eq:npmf} in the subset of\/ $\mathbb{R}_{>0}^\ell$ defined by $h_i(y)>0$ for
$i=1,\dotsc,n$.
\end{theorem}
\begin{proof}
The first statement concerning complex solutions is immediate from
Proposition~\ref{prop:npGale} and the observation that the system~\eqref{Eq:graph_pullback} is
the pullback of the intersection
$\varphi(\mathbb{T}^n)\cap H$ along the
map $\varphi$, if we know that the map $\varphi$ is injective.
Since ${\mathcal{A}}$ spans $\mathbb{Z}^{n}$, so does ${\mathcal{V}}\cup{\mathcal{W}}$, and the map
$\varphi$ is injective.
As the affine span of ${\mathcal{A}}$ has odd index in $\mathbb{Z}^{n}$, the map
$\varphi$ is injective on $(\mathbb{R}^\times)^{n}$.
As $\mathbb{Z}{\mathcal{B}}$ has odd index in its saturation, the equations
$y^\beta\cdot z^\gamma=1$ for $(\beta,\gamma)\in{\mathcal{B}}$ define the image
$\varphi((\mathbb{R}^\times)^{n})$ in the real torus
$(\mathbb{R}^\times)^\ell\times(\mathbb{R}^\times)^n$.
These facts in turn imply the second statement.
Similarly, the hypotheses of the third statement imply the same facts about the
positive part of the real torus, $\mathbb{R}_{>0}^{n}$.
Observing that the subset of $\mathbb{R}_{>0}^\ell$ defined by $h_i(y)>0$ for $i=1,\dotsc,n$ is the
pullback of $\mathbb{R}_{>0}^\ell\times\mathbb{R}_{>0}^n$ under the map $\Psi$ completes the proof.
\end{proof}
\section{Bounds for $(d,\ell)$-dense Gale systems}\label{S:bounds}
By Theorem~\ref{Th:GaleDual}, Theorem~\ref{Th:Structured_bound} follows from
bounds for $(d,\ell)$-dense Gale systems, which we give and prove below.
Let $d,\ell,n$ be positive integers and $y_1,\dots,y_\ell$ be indeterminates.
Suppose that $h_i(y)$ for $i=1,\dots,n$ are generic degree $d$ polynomials.
Define
\begin{eqnarray*}
\dcol{\Delta}&:=&\{y\in\mathbb{R}^\ell_{>0}\mid h_i(y)>0\ \mbox{for}\ i=1,\dots,n\}\,\qquad\mbox{and}\\
\dcol{{\mathcal{M}}(\mathbb{R})}&:=&\{y\in(\mathbb{R}^\times)^\ell\mid h_i(y)\neq0\ \mbox{for}\ i=1,\dots,n\}\,,
\end{eqnarray*}
and write ${\mathcal{M}}(\mathbb{C})$ for the complexification of ${\mathcal{M}}(\mathbb{R})$.
\begin{theorem}\label{TH:main}
With these definitions, suppose that
$\beta_1,\dotsc,\beta_\ell\in\mathbb{Z}^\ell$ and $\gamma_1,\dotsc,\gamma_\ell\in\mathbb{Z}^n$
are vectors such that
${\mathcal{B}}:=\{(\beta_1,\gamma_1),\dotsc,(\beta_\ell,\gamma_\ell)\}\subset\mathbb{Z}^\ell\oplus\mathbb{Z}^n$
are linearly independent.
Then the number of solutions to
\begin{equation}\label{Eq:Gale_Sy}
y^{\beta_j}\cdot h(y)^{\gamma_{j}}\ =\ 1\,,
\qquad\mbox{for}\quad j\ =\ 1,\dotsc,\ell\,,
\end{equation}
in the positive region $\Delta$ is less than
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
If the integer span of ${\mathcal{B}}$ has odd index in its saturation,
then the number of solutions in ${\mathcal{M}}(\mathbb{R})$ is less
than
\[
\frac{e^4{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
\end{theorem}
We will deduce these bounds from several lemmata which we now formulate.
Their proofs are given in subsequent subsections.
For a vector $\alpha$, let $\alpha^\pm$ be the coordinatewise maximum of $\pm\alpha$ and $0$ so
that $\alpha^\pm$ is nonnegative, and $\alpha=\alpha^+-\alpha^-$.
Hence $(1,-2)^+=(1,0)$ and $(1,-2)^-=(0,2)$.
Set
\begin{equation}\label{Eq:form_g}
\dcol{g_k(y)}\ :=\ y^{2\beta_k^+}h(y)^{2\gamma_k^+}-y^{2\beta_k^-}h(y)^{2\gamma_k^-}\,.
\end{equation}
Then $g_k(y)=0$ for $y\in{\mathcal{M}}(\mathbb{C})$ if and only if
\begin{equation}\label{Eq:form_prod}
\prod_{i=1}^\ell (y_i^2)^{\beta_{k,i}}\cdot
\prod_{i=1}^n (h_i(y)^2)^{\gamma_{k,i}}\ =\ 1
\end{equation}
Notice that the system
\[
g_1(y)\ =\ g_2(y)\ =\ \dotsb\ =\ g_\ell(y)\ =\ 0\,,
\]
is equivalent in $\Delta$ to the system~\eqref{Eq:Gale_Sy} and in ${\mathcal{M}}(\mathbb{R})$, it
contains the system~\eqref{Eq:Gale_Sy} as a subsystem.
We will bound the number of solutions to this expanded system in $\Delta$ and in ${\mathcal{M}}(\mathbb{R})$ to obtain
our bounds for the system~\eqref{Eq:Gale_Sy} in Theorem~\ref{TH:main}.
We state two important reductions.
\begin{reduction}\label{R:reduction}
It suffices to prove Theorem~$\ref{TH:main}$ under the following additional assumptions.
\begin{enumerate}
\item
For each $j=1,\dotsc,\ell$, the set
$\dcol{\mu_j}\subset{\mathcal{M}}(\mathbb{C})$ defined by the equations
\[
\dcol{y^{2\beta_k}\cdot h(y)^{2\gamma_k}}\ :=\
\prod_{i=1}^\ell (y_i^2)^{\beta_{k,i}}\cdot
\prod_{i=1}^n (h_i(y)^2)^{\gamma_{k,i}}\ =\ 1\qquad
k=1,\dotsc,j\,,
\]
is smooth and has codimension $j$.
This condition holds for all sufficiently generic polynomials $h_i(y)$ of degree $d$.
\item
For each $k=1,\dotsc,\ell$ define
$b_k:= \beta_{k,1}+\dotsb+\beta_{k,\ell} + d(\gamma_{k,1}+\dotsb+\gamma_{k,n})$.
Then every minor of the $\ell\times(1+\ell+n)$ matrix whose $k$th row is
$(-b_k,\beta_k,\gamma_k)$ is nonzero.
\end{enumerate}
\end{reduction}
We establish these reductions in Subsection~\ref{S:reduction}.
Our bounds are based on an induction which comes from
the Khovanskii-Rolle Theorem, or more precisely, the induction is based on a modified form
which was used in~\cite{BS_mixed},
and which ensures that the hypotheses in subsequent lemmata hold.
See~\cite{BS_Khovanskii} and~\cite{BS_mixed} for more discussion.
For $D$ equal to either ${\mathcal{M}}(\mathbb{R})$ or its positive chamber $\Delta$ and $C$ an algebraic curve in
$D$, let \dcol{$\operatorname{ubc}_D(C)$} be the number of noncompact components of $C$.
Write \dcol{$V_D(f_1,\dotsc,f_\ell)$} for the common zeroes in $D$ of functions
$f_1,\dotsc,f_\ell$.
\begin{lemma}[Modified Khovanskii-Rolle Theorem]\label{L:KhRo}
There exist polynomials $G_1,G_2,\dotsc,G_\ell$ where $G_{\ell-j}(y)$ is a generic
polynomial with degree $2^jn\cdot d$ such that the following hold.
\begin{enumerate}
\item The system
\[
g_1(y)\ =\ \dotsb\ =\ g_j(y)\ =\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\]
has only nondegenerate solutions in ${\mathcal{M}}(\mathbb{C})$ and the system
%
\begin{equation}\label{Eq:C_j}
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ =\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\end{equation}
($g_j$ is omitted) defines a smooth curve in ${\mathcal{M}}(\mathbb{C})$.
\item Let $C_j\subset{\mathcal{M}}(\mathbb{R})$ be the solutions to~\eqref{Eq:C_j} in ${\mathcal{M}}(\mathbb{R})$, which is a smooth
real algebraic curve.
For $D$ be equal to either of ${\mathcal{M}}(\mathbb{R})$ or $\Delta$, we have
\[
|V_D(g_1,\dotsc,g_j,G_{j+1},\dotsc,G_\ell)|\
\leq\ |V_D(g_1,\dotsc,g_{j-1},G_j,\dotsc,G_\ell)|+ \operatorname{ubc}_D(C_j)\,.
\]
\end{enumerate}
\end{lemma}
This implies the following estimate.
Let $D$ equal to either of ${\mathcal{M}}(\mathbb{R})$ or $\Delta$.
Then we have
\begin{equation}\label{Eq:Fundamental_Estimate}
|V_D(g_1,\dotsc,g_\ell)|\ \leq\ |V_D(G_1,\dotsc,G_\ell)| +
\operatorname{ubc}_D(C_1)+\dotsb+\operatorname{ubc}_D(C_\ell)\,.
\end{equation}
Our next lemma estimates these quantities.
\begin{lemma}\label{L:estimates}
We have
\begin{enumerate}
\item $V_\Delta(G_1,\dotsc,G_\ell)\leq V_{\mathcal{M}}(G_1,\dotsc,G_\ell)\leq
2^{\binom{\ell}{2}}n^\ell\cdot d^\ell$,
\item \rule{0pt}{20pt}${\displaystyle\operatorname{ubc}_\Delta(C_{j})\leq
\tfrac{1}{2} 2^{\binom{\ell-j}{2}} n^{\ell-j}
\tbinom{1+\ell+n}{j}}\cdot d^{\ell}$.
\item \rule{0pt}{20pt}${\displaystyle\operatorname{ubc}_{\mathcal{M}}(C_{j})\leq
\tfrac{1}{2} 2^{\binom{\ell-j}{2}} 2^j n^{\ell-j}
\tbinom{1+\ell+n}{j}}\cdot d^{\ell}$.
\item \rule{0pt}{20pt}$ 2^{\binom{\ell-j}{2}} n^{\ell-j}\tbinom{1+\ell+n}{j}\leq
\frac{1}{2} \frac{2^j}{j!} \cdot 2^{\binom{\ell}{2}}n^\ell$.
\end{enumerate}
\end{lemma}
Statement (1) follows from Lemma~\ref{L:KhRo} by B\'ezout's Theorem, as
$2^{\binom{\ell}{2}}n^\ell d^\ell$ is the product of the degrees of the polynomials
$G_1,\dotsc,G_n$.
Statement (4) is from the proof of Lemma 3.5 of~\cite{BS_Khovanskii}.
We prove the other statements of Lemma~\ref{L:estimates} in Subsection~\ref{S:estimates}.
Lemma~\ref{L:estimates} and the estimate~\eqref{Eq:Fundamental_Estimate} give us the
estimate
\[ |V_\Delta(g_1,\dotsc,g_\ell)|\ \leq\
2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\ +\ \sum_{j=1}^\ell
\frac{1}{4} \frac{2^{j}}{j!}\cdot 2^{\binom{\ell}{2}}n^\ell\cdot d^\ell
\ \leq\ \Bigl(1\ +\ \frac{1}{4} \sum_{j=1}^\ell \frac{2^j}{j!}\Bigr)
\cdot 2^{\binom{\ell}{2}}n^\ell\cdot d^\ell \,.
\]
The sum is a partial sum of the power series for $e^2-1$,
and so we obtain
\[
|V_\Delta(g_1,\dotsc,g_\ell)|\ <\
\frac{e^2+3}{4}2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\,.
\]
The estimation for $|V_{\mathcal{M}}(g_1,\dotsc,g_\ell)|$ is similar.
Using Lemma~\ref{L:estimates}(3) for $\operatorname{ubc}_{\mathcal{M}}(C_j)$, the corresponding sum is now a
partial sum for $e^4-1$, and so we obtain
\[
|V_{\mathcal{M}}(g_1,\dotsc,g_\ell)|\ <\
\frac{e^4+3}{4}2^{\binom{\ell}{2}}n^\ell\cdot d^\ell\,,
\]
which completes the proof of Theorem~\ref{TH:main}.
\subsection{Proof of reductions}\label{S:reduction}
The Reduction~\ref{R:reduction}(1) will follow from Bertini's Theorem that a general linear
section of a smooth quasi-projective variety is smooth and of the expected dimension.
First, define $\mathbb{G}_j\subset\mathbb{T}^\ell\times\mathbb{T}^n$ to be the subtorus defined by the equations
\[
y^{2\beta_k}z^{2\gamma_k}\ =\ 1\qquad\mbox{for}\quad k=1,\dots,j\,.
\]
As in Section~\ref{S:gale}, let $\Psi\colon\mathbb{C}^\ell\to\mathbb{C}^\ell\times\mathbb{C}^n$ be the map
defined by
\[
\Psi\ \colon\ y\ \longmapsto\ (y,\,h_1(y),\dotsc,h_n(y))\,.
\]
Then $\mu_j=\Psi^{-1}(\mathbb{G}_j\cap \Psi(\mathbb{C}^\ell))$.
Since $\Psi$ is an isomorphism onto its image and $\mathbb{G}_j$ has codimension $j$, it suffices to
show that $\mathbb{G}_j\cap\Psi(\mathbb{C}^\ell)$ is transverse.
But this follows because $\Psi(\mathbb{C}^\ell)$ is the pullback of a linear subspace $L$ along
the map
\begin{eqnarray*}
\mathbb{C}^\ell\times\mathbb{C}^n &\longrightarrow& \mathbb{C}^{\binom{\ell+d}{d}}\times\mathbb{C}^n\\
(y,z)&\longmapsto&((y^\lambda\colon |\lambda|=d),\, z)\,.
\end{eqnarray*}
The linear space $L$ is defined by the coefficients of the polynomials
in a system Gale dual to the system~\eqref{Eq:Gale_Sy}.
Choosing $L$ to be generic, we may apply Bertini's Theorem and deduce that $\mu_j$ is smooth and
of codimension $j$.
We also see that this may be accomplished by choosing the polynomials $h_i(y)$
to be sufficiently generic.
For the second reduction, observe that our equations~\eqref{Eq:form_g} and~\eqref{Eq:form_prod}
are equivalent to
\begin{equation}\label{eq:f_k}
f_k(y)\ :=\
\sum_{m=1}^\ell \beta_{k,m}\log|y_m|\ +\
\sum_{i=1}^n \gamma_{k,i}\log|h_i(y)|\ =\ 0\,,\qquad
k=1,\dotsc,\ell\,,
\end{equation}
in ${\mathcal{M}}(\mathbb{R})$.
We may perturb them by changing the coefficients $\beta_{k,m}$ and $\gamma_{k,i}$ without
increasing their numbers of nondegenerate solutions.
Thus we can satisfy Reduction~\ref{R:reduction}(2) with real exponents.
Since the rational numbers are dense in the real numbers, we may satisfy
Reduction~\ref{R:reduction}(2) with rational exponents.
Finally, by clearing denominators, we may assume the exponents are integral.
\subsection{Proof of Lemma \ref{L:KhRo}}\label{S:LhRo}
We will establish Lemma~\ref{L:KhRo} by downward induction on $j$.
The main step is provided by the Khovanskii-Rolle Theorem, which we present in the
simplified form in which we need it.
\begin{theorem}[Khovanskii-Rolle]
Let $f_1,\dots,f_\ell$ be smooth functions defined on a domain $D\subset\mathbb{R}^\ell$
where
\[
f_1(y)\ =\ f_2(y)\ =\ \dotsb\ =\ f_{\ell-1}(y)\ =\ 0\,,
\]
defines a smooth curve $C$ in $D$.
Let
\[
J\ :=\ J(f_1,\dots,f_{\ell})\ :=\
\det\left( \frac{\partial f_i}{\partial y_j} \right)_{i,j=1,\dots,\ell}
\]
%
be the Jacobian determinants of $f_1,\dotsc,f_\ell$.
If $V_D(f_1,\dotsc,f_{\ell-1},J)$ is finite and if $C$ has finitely many components in $D$,
then $V_D(f_1,\dotsc,f_\ell)$ is finite and we have
\begin{equation}\label{Eq:KhRoStep}
|V_D(f_1,\dots,f_{\ell})|\ \leq\ |V_D(f_1,\dots,f_{\ell-1},J)|+ \operatorname{ubc}_D(C)\,.
\end{equation}
\end{theorem}
To see this, note that on $C$, the Jacobian $J$ is proportional to the evaluation of the
differential of $f_\ell$ on a tangent vector to $C$.
Given two consecutive solutions $a,b$ to $f_\ell=0$ along an arc of $C$,
The Jacobian will have different signs at $a$ and at $b$, and therefore will vanish at least
once between $a$ and $b$.
\[
\begin{picture}(282,130)(-40,0)
\put(0,0){\includegraphics[height=120pt]{figures/KhovRolle.eps}}
\put(-40,20){$C=V_D(f_1,\dotsc,f_{\ell-1})$}
\put(-20,65){$f_\ell=0$}
\put(148,120){$V_D(f_1,\dotsc,f_{\ell-1},J)$}
\put(146,122){\vector(-3,-1){30}}
\put(19.2,110.3){$a$} \put(26.2,109.3){\vector(1,-1){30}}
\put(200,85.5){$b$} \put(198.2,83.7){\vector(-1,-1){30}}
\end{picture}
\]
The estimate~\eqref{Eq:KhRoStep} follows as
compact components of $C$ contain as many arcs connecting zeroes of $f_\ell$ as zeroes of
$f_\ell$, while noncompact components contain one arc fewer.
To deduce Lemma~\ref{L:KhRo}, we will iterate the Khovanskii-Rolle Theorem, showing
that the appropriate Jacobians have the claimed degrees and ensuring that its
hypotheses are satisfied.
Observe that Lemma~\ref{L:KhRo}(1) with $j=\ell$ holds by the assumptions we make in
Reduction~\ref{R:reduction}.
We prove Lemma~\ref{L:KhRo} by downward induction on $j=\ell,\dotsc,1$.
Specifically, we will assume that Statement (1) holds of some $j$ and then
construct a polynomial $G_j$ such that (2) holds for $j$ and (1) holds for $j{-}1$.
To construct the polynomials $G_j$, we replace the rational functions
$g_k(y)$ for $k=1,\dotsc,\ell$ in Lemma~\ref{L:KhRo}(2) by the logarithmic functions
$f_k(y)$ for $k=1,\dotsc,\ell$ defined in~\eqref{eq:f_k}.
We may do this, as
if $y\in{\mathcal{M}}(\mathbb{R})$, then $f_k(y)=0$ if and only if $g_k(y)=0$.
First, we need to determine the degrees of the Jacobians.
\begin{lemma}\label{L:Newton}
Let $1\leq j\leq \ell$ and suppose that $G_{j{+}1},\dotsc,G_\ell$ are polynomials, where $G_i$ has
degree $2^{\ell-i}n\cdot d$, but is otherwise general, for each $i=1,\dotsc,\ell$.
Then
\begin{equation}\label{Eq:expression}
\prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\cdot
J(f_1,\dotsc,f_j,\, G_{j+1},\dotsc,G_\ell)
\end{equation}
is a polynomial with degree $2^{\ell-j}n\cdot d$.
\end{lemma}
We use this to deduce Lemma~\ref{L:KhRo}.
Suppose that we have polynomials $G_{j+1},\dotsc,G_\ell$ where $G_i$ is a generic
polynomial with degree $2^{\ell-i}n\cdot d$, for each $i=j{+}1,\dotsc,\ell$,
and Lemma~\ref{L:KhRo}(1) holds for $j$.
Let $C_j\subset{\mathcal{M}}(\mathbb{R})$ be the smooth real algebraic curve defined by
\[
f_1(y)\ = \ \dotsb\ =\ f_{j-1}(y)\ \;=\;\
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,.
\]
Let $\overline{G}_j$ be the product of the Jacobian
$J(f_1,\dotsc,f_j,\, G_{j{+}1},\dotsc,G_\ell)$ with the polynomial
\[
\Upsilon(y)\ :=\ \prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\,.
\]
Then $\overline{G}_j$ is a polynomial with degree $2^{\ell-j}n\cdot d$, by
Lemma~\ref{L:Newton}.
Since $\Upsilon$ does not vanish in ${\mathcal{M}}(\mathbb{R})$, the polynomial $\overline{G}_\ell$
and the Jacobian define the same sets, and we have
\begin{multline}\label{Eq:KhRo_est}
\qquad|V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)|\ \leq\ \\
|V_D(f_1,\dotsc,f_{j-1},\,\overline{G}_j,\,G_{j+1},\dotsc,G_\ell)|\ +\
\operatorname{ubc}_D(C_j)\,,\qquad
\end{multline}
by the Khovanskii-Rolle Theorem.
Note however that we do not know if $\overline{G}_j$ is a generic polynomial
with degree $2^{\ell-j}n\cdot d$, and in particular, we do not know if the
hypotheses of Lemma~\ref{L:KhRo}(1) for $j{-}1$ hold.
These hypotheses may be achieved by perturbing $\overline{G}_j$ to a nearby generic
polynomial $G_j$ with degree $2^{\ell-j}n\cdot d$.
To ensure that this perturbation does not destroy the estimate~\eqref{Eq:KhRo_est},
we only need to guarantee that the signs of $G_j$ and $\overline{G}_j$ are the same at
every point of $V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)$, but this will hold for all
sufficiently small perturbations, as there are only finitely many such points
and $\overline{G}_j$ is nonzero at each.
These conditions are equivalent to every point of
$V_D(f_1,\dotsc,f_j,\,G_{j+1},\dotsc,G_\ell)$ being nondegenerate, which
is ensured by the genericity of $G_{j{+}1},\dotsc,G_\ell$.
This completes the proof of Lemma~\ref{L:KhRo}.
\begin{proof}[Proof of Lemma~$\ref{L:Newton}$]
Write $\partial_m$ for $\partial/\partial y_m$,
and consider the expression~\eqref{Eq:expression}, writing the Jacobian in block form,
with $j$ rows for $f_1,\dotsc,f_j$ and $\ell-j$ rows for $G_{j+1},\dotsc,G_\ell$.
\begin{multline}\label{Eq:block}
\qquad \prod_{m=1}^\ell y_m\cdot\prod_{i=1}^n h_i(y)\cdot
\det\left(\, \begin{matrix}
\bigl(\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}
\smallskip\\\hline\rule{0pt}{15pt}
\bigl(\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}
\end{matrix}\,\right)\\
\ =\
\prod_{i=1}^n h_i(y)\cdot
\det\left(\, \begin{matrix}
\bigl(y_m\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}
\smallskip\\\hline\rule{0pt}{15pt}
\bigl(y_m\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}
\end{matrix}\,\right)\,.\qquad
\end{multline}
Laplace expansion along the first $j$ rows of the matrix on the right expresses its
determinant as a sum of products of maximal minors of the two blocks.
We will prove Lemma~\ref{L:Newton} by showing that each term in that sum is a polynomial with
degree $2^{\ell-j}n\cdot d$.
First, the lower block
$\bigl(y_m\partial_m G_k(y)\bigr)_{k=j{+}1,\dotsc,\ell}^{m=1,\dotsc,\ell}$
is a matrix of polynomials whose entries in row $k$ are the toric derivatives
$y_m\partial_m G_k(y)$ of $G_k$.
Thus every entry in row $k$ has degree $\deg(G_k)=2^{\ell-k}n\cdot d$, and
therefore each minor has degree
\begin{equation}\label{Eq:lower_degree}
2^{\ell-(j+1)}n\cdot d \ +\ \dotsb\ +\
2n\cdot d \ +\ n\cdot d\ =\
(2^{\ell-j}-1)n\cdot d\,.
\end{equation}
For the upper block, note that $y_m\partial_m f_k(y)$ is
\[
\beta_{k,m}\ +\ \sum_{i=1}^k \gamma_{k,i}y_m\partial_m \log|h_i(y)|\ =\
\beta_{k,m}\ +\ \sum_{i=1}^n\gamma_{k,i} \frac{y_m \partial_m h_i(y)}{h_i(y)}\,.
\]
In particular, the upper block is a product of a $j\times(\ell+n)$ matrix and a
$(\ell+n)\times \ell$ matrix,
\[
\bigl(y_m\partial_m f_k(y)\bigr)_{k=1,\dotsc,j}^{m=1,\dotsc,\ell}\ =\
\bigl(\, \beta_{k,q}\ \mid\ \gamma_{k,i}\,\bigr)\cdot
\left(\begin{matrix} I_\ell\medskip\\
y_m \partial_m h_i(y)/h_i(y)\end{matrix}\right)\ ,
\]
where $I_\ell=(\delta_{q,m})$ is the $\ell\times\ell$ identity matrix.
By the Cauchy-Binet formula, a $j\times j$ minor of $\bigl(y_m\partial_m f_k(y)\bigr)$ is a
sum of products of $j\times j$ minors of the two matrices on the right.
Consider now the product of $\prod_{i=1}^nh_i(y)$ with a term in this sum.
The first matrix $\bigl(\beta_{k,q}\, \mid\, \gamma_{k,i}\bigr)$ contains constants, and
a $j\times j$ minor of the second involves no more than $p:=\min\{n,j\}$ rows from
its lower $n\times\ell$ block $(y_m\partial_mh_i(y)/h_i(y))$.
This minor is a sum of $j!$ terms, each one a product of a constant and $p$ entries of the
matrix of the form $y_m\partial_mh_i(y)/h_i(y)$, for different rows $i$.
Multiplying this term by $\prod_{i=1}^nh_i(y)$ will clear all
denominators and result in a product of $p$ terms of the form
$y_m\partial_mh_i(y)$ and complementary $n{-}p$ terms of the form $h_i(y)$.
Each of these terms has degree $d$, so each term coming from the
expansion of this $j\times j$ minor has degree $n\cdot d$, and therefore
the product of $\prod_{i=1}^nh_i(y)$ by each $j\times j$ minor of the upper block
of~\eqref{Eq:block} will have degree $n\cdot d$.
Together with~\eqref{Eq:lower_degree}, this completes the proof.
\end{proof}
\subsection{Proof of Lemma~\ref{L:estimates}}\label{S:estimates}
We only need to prove Statements (2) and (3) of Lemma~\ref{L:estimates}.
By Reduction~\ref{R:reduction} and Lemma~\ref{L:KhRo}, we may assume that the polynomials
$h_i(y)$ and $G_j(y)$ are generic given their degrees.
The complexification of the real curve $C_j$ is defined in $\mathbb{C}^\ell\supset{\mathcal{M}}(\mathbb{C})$ by
\begin{equation}\label{Eq:curvej}
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ \ =\ \
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,,
\end{equation}
and it lies on the codimension $j{-}1$ subvariety $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$ defined by
\[
g_1(y)\ =\ \dotsb\ =\ g_{j-1}(y)\ =\ 0\,.
\]
We bound the number of unbounded components of $C_j$ by first describing the points where
$\mu_{j-1}(\mathbb{C})$ meets the boundary of ${\mathcal{M}}(\mathbb{C})$, then bound the number of real solutions to
\begin{equation}\label{Eq:bigG}
G_{j+1}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0
\end{equation}
on these boundary points, and lastly by determining the number of components of $C_j$ incident
upon each such real solution.
To accomplish this, consider ${\mathcal{M}}(\mathbb{C})\subset\mathbb{C}^\ell$ as a subset of projective space
$\P^\ell$.
Its boundary $\partial{\mathcal{M}}(\mathbb{C}):=\P^\ell\setminus{\mathcal{M}}(\mathbb{C})$
consists of the finite coordinate planes $y_m=0$ for $m=1,\dotsc,\ell$, the coordinate
plane at inifinty $y_0=0$, and the degree $d$ hypersurfaces $h_i(y)=0$ for $i=1,\dotsc,n$.
Strictly speaking, we must homogenize polynomials $h_i,g_j,G_k$ with respect to the coordinate
$y_0$ at infinity.
When working on an affine patch where $y_m\neq 0$, we de-homogenize them by setting
$y_m=1$.
By our assumption that the polynomials $h_i$ were general, this boundary
$\partial{\mathcal{M}}(\mathbb{C})$ forms a divisor with normal crossings whose components are the coordinate
planes and the hypersurfaces $h_i(y)=0$.
The common zeroes of any $q$ of the polynomials $h_1,\dotsc,h_n$ and $j{-}q$ of the
coordinates $y_0,\dotsc,y_\ell$ is a smooth subvariety of codimension $j$, called a
\demph{codimension-$j$ stratum}.
The union of these $j$-fold intersections of the components of the boundary divisor
is called the \demph{codimension-$j$ skeleton} of $\partial{\mathcal{M}}(\mathbb{C})$.
\begin{lemma}
The closure $\overline{\mu_{j-1}}$ meets $\partial{\mathcal{M}}(\mathbb{C})$ in a union of
codimension-$j$ strata and in the neighborhood of a real point of\/
$\overline{\mu_{j-1}}$ lying in the relative interior of a codimension-$j$ stratum,
$\mu_{j-1}$ has one branch in each of the $2^j$ components of ${\mathcal{M}}(\mathbb{R})$
incident on that point.
\end{lemma}
\begin{proof}
Since $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$ has codimension $j{-}1$ in $\P^\ell$, the
intersection of its closure $\overline{\mu_{j-1}}$ with the
boundary divisor $\partial{\mathcal{M}}(\mathbb{C})$ will have codimension $j$ in $\P^\ell$.
We prove the first part of the lemma by showing that this intersection consists of points
lying within the codimension-$j$ skeleton of $\partial{\mathcal{M}}(\mathbb{C})$, and therefore is a union of
components of the codimension-$j$ skeleton.
Let $Y$ be a point of $\partial{\mathcal{M}}(\mathbb{C})$ that does not lie in the codimension-$j$ skeleton.
We show that $Y\not\in\overline{\mu_{j-1}}$.
Since $\partial{\mathcal{M}}(\mathbb{C})$ is a divisor with normal crossings whose components are defined by
the coordinates $y_i$ and forms $h_i$, at least one, but no more than
$j{-}1$ of the coordinates $y_0,y_1,\dotsc,y_\ell$ and forms
$h_1,\dotsc,h_n$ vanish at $Y$.
Reordering the coordinates and forms if necessary, we may assume that the forms which
vanish at $Y$ are among $h_1,\dotsc,h_q$ and the coordinates are among
$y_{q+1},\dotsc,y_{j-1}$.
Since the assertion about $Y$ is local, we may restrict to the affine patch $U$ where
none of the remaining coordinates or forms vanish.
The equations
\begin{equation}\label{Eq:mujeqs}
y^{2\beta_k} h(y)^{2\gamma_k}\ =\ 1\qquad k=1,\dotsc,j{-}1\,,
\end{equation}
define $\mu_{j-1}\subset{\mathcal{M}}(\mathbb{C})$.
By Reduction~\ref{R:reduction}(2) on the homogenized exponent vectors,
there is an integer linear combination of the first $j{-}1$
rows of the matrix $(-b_k,\beta_k,\gamma_k)$ so that the columns corresponding to
$h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_{j-1}$ become diagonal.
These same linear combinations transform the equations~\eqref{Eq:mujeqs} into equations
of the form
\begin{equation}\label{Eq:skeletondiagonal}
\begin{array}{rcl}
h_i(y)^{a_i} &=& y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad i=1,\dotsc,q\,,\\
y_i^{a_i} &=& y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad
i=q{+}1,\dotsc,j{-}1\,,\rule{0pt}{15pt}
\end{array}
\end{equation}
where $a_i> 0$ and the components of $\alpha_i$ in positions $q{+}1,\dotsc,j{-}1$
vanish as do the components of $\delta_i$ in positions $1,\dotsc,q$.
That is, $h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_{j-1}$ do not appear on the right of these
expressions.
Since the expressions in~\eqref{Eq:skeletondiagonal} are well-defined functions in $U$,
the regular functions
\[
\begin{array}{rl}
h_i(y)^{a_i}\ -\ y^{\alpha_i}\cdot h(y)^{\delta_i}\quad &\mbox{for}\quad i=1,\dotsc,q\,,
\quad\mbox{and}\\
y_i^{a_i}\ -\ y^{\alpha_i}\cdot h(y)^{\delta_i}\quad&
\mbox{for}\quad i=q{+}1,\dotsc,j{-}1\rule{0pt}{15pt}
\end{array}
\]
vanish on $\mu_{j-1}\cap U$, and hence on $\overline{\mu_{j-1}}\cap U$.
However, these cannot all vanish at $Y$, for none of the functions
$y^{\alpha_i}h^{\delta_i}$ for $i=1,\dotsc,j{-}1$ vanish at $Y$, but at least one of
$h_1^{a_1},\dotsc,h_q^{a_q}$, $y_{q+1}^{a_{q+1}},\dotsc,y_{j-1}^{a_{j-1}}$ vanishes at
$Y$.
To complete the proof, suppose that $Y\in\overline{\mu_{j-1}}(\mathbb{R})\cap\partial{\mathcal{M}}(\mathbb{R})$ is a
real point lying on a
codimension-$j$ stratum of $\partial{\mathcal{M}}(\mathbb{C})$ but not on a stratum of larger codimension.
Reordering the coordinates and functions if necessary and working locally, we may assume that
the polynomials $h_1(y),\dotsc,h_q(y)$ and coordinates $y_{q+1},\dotsc,y_{j}$ vanish at
$Y$.
Thus in the affine neighborhood $U$ of $Y$ where none of the other polynomials or coordinates
vanish, $\mu_{j-1}$ is defined by equations of the form
\begin{equation}\label{Eq:newdiagonal}
\begin{array}{rcl}
h_i(y)^{a_i} &=& y_j^{c_i}\cdot y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad i=1,\dotsc,q\,,\\
y_i^{a_i} &=& y_j^{c_i}\cdot y^{\alpha_i}\cdot h(y)^{\delta_i}\qquad
i=q{+}1,\dotsc,j\,,\rule{0pt}{15pt}
\end{array}
\end{equation}
where $a_i>0$ and none of $h_1,\dotsc,h_q$ and $y_{q+1},\dotsc,y_j$ appear in the expressions
$y^{\alpha_i}\cdot h(y)^{\delta_i}$ for $i=1,\dotsc,j{-}1$.
In fact, we must have $c_i>0$ as $Y\in\overline{\mu_{j-1}}$.
In a neighborhood of $Y$ in $\mathbb{R}\P^\ell$, the complement ${\mathcal{M}}(\mathbb{R})$ has $2^j$ chambers given
by the signs of the functions $h_1(y),\dotsc,h_q(y)$, $y_{q+1},\dotsc,y_j$.
Since the exponents in~\eqref{Eq:newdiagonal} have every component even
(this comes from the evenness of the exponents in~\eqref{Eq:mujeqs}), there is a
component of $\mu_{j-1}$ in each of these chambers.
\end{proof}
We complete the proof of Statements (2) and (3) of Lemma~\ref{L:estimates}.
We estimate the number of unbounded components of the curve $C_j$ by first bounding the
number of points where its closure in $\mathbb{R}\P^\ell$ meets the boundary divisor, and then
bounding the number of components of $C_j$ incident upon each point.
The estimate for the number of points in $\overline{C_j}\cap\partial{\mathcal{M}}(\mathbb{R})$ is simply
the number of points in the codimension-$j$ skeleton where
\begin{equation}\label{eq:Gs}
G_{j+1}(y)\ =\ G_{j+2}(y)\ =\ \dotsb\ =\ G_\ell(y)\ =\ 0\,.
\end{equation}
Consider a stratum where $q$ of the polynomials $h_i$ vanish and $j{-}q$ of the coordinates
$y_m$ vanish.
Since the polynomials $G_i$ and $h_i$ are general given their degrees, the number of
points on this stratum will be at most the product of these degrees, which is
\[
d^q \cdot 2^{\ell-j-1} nd\cdot 2^{\ell-j-2} nd \dotsb 2 nd \cdot nd \ =\
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} d^{\ell-j} d^q\,.
\]
Since there are $\binom{\ell+1}{j-q}\cdot\binom{n}{q}$ such strata, the number of points
where $C_j$ meets the boundary is at most
\begin{equation}\label{eq:cjbdry}
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} d^{\ell-j}
\cdot\ \sum_{q=0}^j \binom{\ell+1}{j-q}\binom{n}{q}\cdot d^q\ <\
2^{\binom{\ell-j}{2}} \cdot n^{\ell-j} \binom{1+\ell+n}{j}
\cdot d^\ell\ .
\end{equation}
As the polynomials $G_i$ are general the variety defined by~\eqref{eq:Gs} is transverse to
the codimension-$j$ stratum, so there is at most one branch of $C_j$ in each component of
${\mathcal{M}}(\mathbb{R})$ meeting such a point.
Thus the number~\eqref{eq:cjbdry} bounds the number of ends of components
of $C_j$ in $\Delta$, so it bounds twice the number of unbounded components of $C_j$ in
$\Delta$.
For the bound in ${\mathcal{M}}(\mathbb{R})$, we multiply this by $2^j$, as there are $2^j$ components of
${\mathcal{M}}(\mathbb{R})$ meeting each such point, and each component of ${\mathcal{M}}(\mathbb{R})$ contains at most one branch of
$C_j$.
This completes the proof of the Lemma~\ref{L:estimates}.
\section{Betti Number Bounds}\label{Sec:Betti}
Using the bound~\eqref{Eq:structured_bound} and stratified Morse theory for
a manifold with corners~\cite{GM88}, we prove the following theorem.
\begin{theorem}\label{betti}
Let $X$ be a hypersurface in $\mathbb{R}^n_>$ defined by a $(d,\ell)$-dense fewnomial.
Then
\[
b_*(X)\ <\ \frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
Let $f(x_1,\dots,x_n)$ be a real Laurent polynomial with $(d,\ell)$-dense support
${\mathcal{A}} \subset \mathbb{Z}^n$ such that
$X := {\mathcal{V}}(f) \subset \mathbb{R}^n_>$ is a smooth hypersurface.
By a logarithmic change of coordinates, $x_i=e^{z_i}$, we may work with exponential sums in
$\mathbb{R}^n$ instead of polynomials in $\mathbb{R}^n_>$.
Then the $(d,\ell)$-dense fewnomial
$f = \sum_{\alpha \in {\mathcal{A}}} c_{\alpha} x^{\alpha}$ becomes
the exponential sum
\[
\varphi\ :=\ \sum_{\alpha \in {\mathcal{A}}} c_{\alpha} e^{z \cdot \alpha}.
\]
In this way, the bounds~\eqref{Eq:BS} and~\eqref{Eq:structured_bound} for positive
solutions to fewnomial systems hold for real solutions to systems of exponential sums with the
same exponents.
Let $Z := {\mathcal{V}}(\varphi) \subset \mathbb{R}^n$ be the hypersurface defined by $\varphi$, which is
homeomorphic to $X$. We will prove Theorem~\ref{betti} in these logarithmic coordinates
and with real exponents.
\begin{theorem}\label{bettilog}
The sum of the Betti numbers of a hypersurface in $\mathbb{R}^n$ defined by an exponential sum
whose exponent vectors are $(d,\ell)$-dense
${\mathcal{A}} = {\mathcal{W}}\, \cup\, \psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell) \subset \mathbb{R}^n$ is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
An affine change of coordinates replaces ${\mathcal{W}}$ with another
set of independent vectors and replaces $\psi$ with another affine map but it
does not change the $(d,\ell)$-dense structure.
We may thus assume that the vectors in ${\mathcal{W}}$ are the standard unit basis vectors
in $\mathbb{R}^n$, and so $\varphi$ includes the coordinate exponentials $e^{z_i}$ for
$i = 1,\dots,n$.
Let $M \colon = (M_0,M_1,\dots,M_n)$ be a list of positive numbers and set
\[
\Delta(M) \ :=\ \{z \in \mathbb{R}^n \mid z_i \geq -M_i,\, i = 1,\dots,n \quad \text{ and }
\quad \sum_i z_i \leq M_0\},
\]
which is a nonempty simplex. We will use stratified Morse theory to bound the Betti
numbers of $Y := Z \cap \Delta_M$ when $M$ is general.
\begin{theorem}\label{bettigeneral}
For $M$ general, the sum of the Betti numbers of\/ $Y$ is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell
\cdot \sum_{i=0}^{n} \tbinom{n}{i}\, i^\ell\,.
\]
\end{theorem}
Theorem~\ref{bettilog} is a consequence of Theorem~\ref{bettigeneral}.
See Theorem $1'$ in~\cite{BS_Betti} for a detailed proof.
\begin{proof}[Proof of Theorem~$\ref{bettigeneral}$]
Given positive numbers $M = (M_0,M_1,\dots,M_n)$, define affine hyperplanes in $\mathbb{R}^n$
by
\[
H_0\ :=\ \{z \mid \sum_i z_i = M_0\} \quad \text{ and } \quad
H_i\ :=\ \{z \mid z_i = -M_i\}, \quad \text{ for } i = 1,\dots,n\,.
\]
For each proper subset $S \subset \{0,\dots,n\}$, define an affine linear subspace
\[
H_S\ :=\ \bigcap_{i \in S} H_i\,.
\]
Since each $M_i > 0$, this has dimension $n-|S|$, and these subspaces are the affine
linear subspaces supporting the faces of the simplex $\Delta_M$.
Choose $M$ generic so that for all $S$ the subspace $H_S$ meets $Z$ transversally.
For each subset $S$, set $Z_S := Z \cap H_S$. This is a smooth hypersurface in $H_S$ and
therefore has dimension $n-|S|-1$. The boundary stratum $Y_S$ of $Y = Z \cap \Delta_M$
lying in the relative interior of the face supported by $H_S$ is an open subset of $Z_S$.
For a nonzero vector $u \in \mathbb{R}^n$, the directional derivative $D_u\varphi$ is
\[
\sum_{\alpha \in {\mathcal{A}}} (u \cdot \alpha)c_{\alpha}e^{z \cdot \alpha}
\]
which is an exponential sum having the same exponents as $\varphi$. Let $L_u$ be the
linear function on $\mathbb{R}^n$ defined by $z \mapsto u \cdot z$.
The critical points of the function $L_u$ restricted to $Z$ are the zeroes of the
system
\[
\varphi(z) = 0 \quad \text{ and } \quad
D_v \varphi(z) = 0 \quad \text{ for } v \in u^{\perp}\,.
\]
When $u$ is general and we choose a basis for $u^\perp$, this becomes a system of
$n$ exponential sums in $n$ variables having the same support as the original polynomial.
Therefore, the whole system has $(d,\ell)$-dense support,
${\mathcal{A}}=\psi(d\includegraphics{figures/Simplex.eps}^\ell\cap\mathbb{Z}^\ell)\cup \{e_1,\dotsc,e_n\}$.
By Theorem~\ref{Th:Structured_bound}, the number of solutions is at most
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, n^\ell \cdot d^\ell\,.
\]
We use this to estimate the number of critical points of the function $L_u$ restricted
to $Z_S$.
The restriction $\varphi_S$ of $\varphi$ to $H_S$ defines $Z_S$ as a hypersurface in $H_S$.
We determine this restriction.
Suppose first that $0\notin S$.
If $i\in S$, then we may use the equation $z_i = -M_i$ to eliminate the variable $z_i$ and
the exponential $e^{z_i}$ from $\varphi$.
The effect of these substitutions for $i\in S$ on the exponents is the projection $\pi$
sending $e_i\mapsto 0$ for $i\in S$.
Then $\pi \circ \psi$ is still affine and so
$\varphi_S$ is still a $(d,\ell)$-dense fewnomial but with $n$ replaced by $n{-}|S|$,
and thus the number of critical points of $L_u|_{H_S}$ on $Z_S$ is bounded by
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}}\, (n-|S|)^\ell \cdot d^\ell\,.
\]
If $0 \in S$, then we could use fewnomial theory to estimate the number of critical
points of $L_u|_{H_S}$ on $Z_S$, but will not need that estimate.
Let $u$ be a general vector in $\mathbb{R}^n$ such that $L_u$ is a Morse function for the
stratified space $Y$. By Proposition $2$ in~\cite{BS_Betti}, the sum of the Betti numbers
of $Y$ is bounded by the number of critical points $p$ of $L_u$ for which $L_u$ achieves
its minimum on the normal slice $N(p)$ at $p$. Since the strata $Y_S$ of $Y$ are open
subsets of the manifolds $Z_S$, this number is bounded above by the number of such
critical points of $L_u$ on the manifolds $Z_S$. Just as in~\cite{BS_Betti}, we can alter
$u$ so that no critical point in any $Z_S$ with $0 \in S$ contributes. Therefore,
the sum of the Betti numbers of $Y$ is bounded above by
\[
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell \cdot
\sum_{S \subset \{1,\dots,n\}} (n-|S|)^\ell
\ =\
\frac{e^2{+}3}{4}\, 2^{\binom{\ell}{2}} \cdot d^\ell \cdot
\sum_{i=0}^{n} {\binom{n}{i}} (n-i)^\ell\,.\vspace{-30pt}
\]
\end{proof}\vspace{15pt}
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\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
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| train/arxiv |
BkiUbbM5qsNCPSsgaKqk | 5 | 1 | \section{Introduction}
Since the pioneering Bell paper \cite{Bell} the Bell inequalities became the subject of intensive study (for a review see \cite{Liang},\cite{Brunner}). Their importance stems from the fact that their violation at the quantum level provides the evidence that the quantum theory cannot be viewed as a local realistic theory. Another important notion in physics is that of symmetry. On the formal level various symmetries are described in the framework of group theory. Therefore, it appears natural to study Bell inequalities for the systems described by the sets of states classified by the representations of some groups. This idea has been proposed in the interesting papers by G\"uney and Hillery \cite{Guney},\cite{Guney1} and studied in some detail by the present authors \cite{Bolonek,Bolonek1,Bolonek2,Bolonek3}. Within this approach the states of quantum system entering some Bell inequality form the orbit(s) of a particular unitary representation of some group $G$. The orbit is chosen in such a way as to consist of a disjoint sum of subsets, each forming an orthonormal basis in the space of states. Each such basis provides the spectral decomposition of some observable. Grouptheoretical methods allow for simple calculation of quantum bound on the combination of probabilities entering the particular Bell's inequality. Computing classical bound calls for more effort. However, one can use here Fine's theorem \cite{Fine} which considerably reduces the relevant combinatorics.
Recently, Tavakoli and Gisin \cite{Tavakoli} in the very nice paper constructed Bell inequalities which violations are attained with measurements pointing to the vertices of the Platonic and Archimedean solids. Again we are dealing with the systems exhibiting high degree of symmetry; it is the symmetry which makes the Platonic solids so beautiful. In the present paper we generalize the Tavakoli \& Gisin approach using the tools from elementary group theory. Our starting point is the idea to look at the Platonic and Archimedean solids (as well as some other threedimensional objects) as generic and nongeneric orbits of threedimensional representations of some finite groups (basically, the subgroups of relevant symmetry groups). It appears that this point of view brings some advantages. We obtain an extremally simple formula for quantum value of the combination of correlation functions introduced in \cite{Tavakoli} under the assumption that our quantum system is described by the maximally entangled $SO(3)$ singlet state. We find also nice geometric picture, following from Fine's theorem and group theory, of classical states saturating Bell inequality. A number of examples of considerable violation of Bell inequality is presented.
\section{The general scheme}
We consider a Bell experiment with Alice and Bob having a number of possible settings characterized by the sets of directions in space, $\poisson{\vec{v}_i}_{i=1}^{N_A}$ and $\poisson{\vec{w}_j}_{j=1}^{N_B}$, respectively. They measure the spin $-\frac{1}{2}$ projections in the $\vec{v}_i$ (Alice) and $\vec{w}_j$ (Bob) directions; the corresponding observables $A_i$ and $B_j$ acquire the values $\pm 1$ depending on whether the projection is positive or negative.
One expects the Bell inequality to be violated the more the larger is the degree of entanglement of the state of the system. Therefore, we choose this state to be the $SO(3)$ singlet
\begin{equation}
\ket{\phi}\equiv\frac{1}{\sqrt{2}}\naw{\ket{\downarrow\uparrow}-\ket{\uparrow\downarrow}}.\label{1}
\end{equation}
It differs from the state $\ket{\phi^+}\equiv\frac{1}{\sqrt{2}}\naw{\ket{\uparrow\uparrow}+\ket{\downarrow\downarrow}}$ used by Tavakoli and Gisin \cite{Tavakoli}; however it is maximally entangled, $\mathrm{Tr}_A\naw{\ket{\phi}\bra{\phi}}=\frac{1}{2}\mathbbm{1}=\mathrm{Tr}_B\naw{\ket{\phi}\bra{\phi}}$ and slightly more convenient from the theoretical point of view since the relevant correlation functions are rotationally invariant. Moreover, most results obtained for $\ket{\phi^+}$ remain valid here (see below).
Simple calculation yields the well-known formula for the correlation function
\begin{equation}
\av{A_iB_j}_\phi\equiv \bra{\phi}\naw{\vec{v}_i\cdot\vec{\sigma}}\otimes\naw{\vec{w}_j\cdot\vec{\sigma}}\ket{\phi}=-\vec{v}_i\cdot\vec{w_j} \label{2}
\end{equation}
which is rotationally invariant as expected.
The Bell inequality we are considering concerns a linear combination of correlation functions
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}c_{ij}\av{A_iB_j}.\label{3}
\end{equation}
Following \cite{Tavakoli} we adjust the coefficients $c_{ij}$ in such a way as to optimize the quantum bound on $\mathcal{B}$. Therefore, we put
\begin{equation}
c_{ij}=-\vec{v}_i\cdot\vec{w}_j\label{4}
\end{equation}
and eq.~(\ref{3}) takes the form
\begin{equation}
\mathcal{B}=-\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\vec{w}_j}\av{A_iB_j}.\label{5}
\end{equation}
In particular, for the singlet state (\ref{1})
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\vec{w}_j}^2.\label{6}
\end{equation}
In Ref.~\cite{Tavakoli} the directions $\vec{v}_i$, $\vec{w}_j$ correspond to the vertices of Platonic solids (and Archimedean ones). Therefore, they enjoy a high degree of symmetry. In particular, the most symmetric case corresponds to dual Platonic solids determining Alice and Bob settings. Dual Platonic solids share the same symmetry group. Guided by this example we consider the following general situation. Let $G=\poisson{g_\alpha}_{\alpha=1}^{|G|}$ $(g_1=e)$ be some finite group admitting threedimensional real irreducible representation; some reducible representations can be also considered (see below). We assume that $\poisson{v_i}_{i=1}^{N_A}$, $\poisson{w_j}_{j=1}^{N_B}$ are two arbitrary orbits of $G$, generated by the representation $D(g)$ under consideration. The orbits of $G$ are obtained by acting with all matrices $D(g)$ on some fixed vectors,
\begin{equation}
\vec{v}_\alpha=D(g_\alpha)\vec{v}\label{7}
\end{equation}
\begin{equation}
\vec{w}_\beta=D(g_\beta)\vec{w}.\label{8}
\end{equation}
For generic $\vec{v}$ ($\vec{w}$) we get $|G|$ vectors $\vec{v}_\alpha$ ($\vec{w}_\beta$). However, it can happen that the stability subgroup of $\vec{v}$ ($\vec{w}$), $G_v$ ($G_w$) is nontrivial. Then the vectors $\vec{v}_\alpha$ ($\vec{w}_\beta$) are in one-to-one correspondence with the cosets in $^G/_{G_v}$ ($^G/_{G_w}$), i.e. $\vec{v}_\alpha=\vec{v}_{\alpha'}$ ($\vec{w}_\beta=\vec{w}_{\beta'}$) iff $g_\alpha$, $g_{\alpha'}$ ($g_\beta, g_{\beta'}$) belong to the same coset of $^G/_{G_v}$ ($^G/_{G_w}$). Note that $^{|G|}/_{|G_v|}=N_A$, $^{|G|}/_{|G_w|}=N_B$.
Consider now the following sum
\begin{equation}
\begin{split}
&\sum_{g,g'\in G}\naw{D(g')\vec{v}\cdot D(g)\vec{w}}^2=\sum_{g,g'\in G}\naw{\vec{v}\cdot D(g'^{-1})\cdot D(g)\vec{w}}^2\\
&=\sum_{g,g'\in G}\naw{\vec{v}\cdot D((g'^{-1}\cdot g))\vec{w}}^2=|G|\sum_{g\in G}\naw{\vec{v}\cdot D(g)\vec{w}}^2\\
&=|G|\sum_{a,b,c,d=1}^3\sum_{g\in G}v_a w_b v_c w_d D_{ab}(g) D_{cd}(g)
\end{split} \label{9}
\end{equation}
where we have changed the summation indices, $g'\rightarrow g'$, $g'^{-1}\cdot g\rightarrow g$.
By applying the orthogonality relations
\begin{equation}
\sum_{g\in G}\overline{D_{ab}^{(\mu)}(g)}D_{cd}^{(\nu)}(g)=\frac{|G|}{\dim D^{(\mu)}}\delta_{\mu\nu}\delta_{ac}\delta_{bd}\label{10}
\end{equation}
and the normalization conditions $|\vec{v}|=|\vec{w}|=1$ we find
\begin{equation}
\sum_{g,g'\in G}\naw{D(g')\vec{v}\cdot D(g)\vec{w}}^2=\frac{1}{3}|G|^2.\label{11}
\end{equation}
Now, any term of the sum defining $\mathcal{B}$, eq.~(\ref{6}), enters the left-hand side of (\ref{11}) $|G_{v}|\cdot|G_w|$ times. Therefore, one finds finally the following expression for $\mathcal{B}$ for the maximally entangled state (\ref{1})
\begin{equation}
\mathcal{B}=\frac{1}{3}N_A\cdot N_B. \label{12}
\end{equation}
This is an extremally simple formula in terms of the Alice and Bob numbers of settings. It is important to point out that the only assumption we made is that $G$ generates the orbits of Alice and Bob; $G$ does not have to be the full symetry group of either orbit; the latter is much stronger assumption. For example, the symmetric group $S_4$ is the symmetry group of the regular tetrahedron which is some nongeneric orbit of threedimensional representation of $S_4$; however, for judicious choice of initial vector one can also obtain octahedron which has larger symmetry group $S_4\times S_2$.
Consider now the classical bound $C$ on $\mathcal{B}$. Its determination is slightly more involved. According to Fine's theorem \cite{Fine} it is sufficient to compute $\mathcal{B}$, eq.~(\ref{6}), for all deterministic responses of Alice and Bob and pick the largest one \cite{Tavakoli}, \cite{Bolonek}.
Therefore, we obtain \cite{Tavakoli}
\begin{equation}
\begin{split}
&\mathcal{B}\leq C\equiv \max\limits_{\substack{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}\\ B_1,\ldots,B_{N_B}\in\{\pm 1\}^{N_B}}}\naw{-\sum_{j=1}^{N_B}B_j \sum_{i=1}^{N_A}A_i\naw{\vec{v}_i\cdot\vec{w}_j}}\\
&=\max\limits_{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}}\naw{\sum_{j=1}^{N_B}\modu{\sum_{i=1}^{N_A}A_i\naw{\vec{v}_i\cdot\vec{w}_j}}}.\end{split}\label{13}
\end{equation}
The bound (\ref{13}) can be effectively computed provided $N_A$ and $N_B$ are not too large.
In order to get more insight into the structure of classical bound let us rewrite $C$ in the form
\begin{equation}
C=\max\limits_{\substack{A_1,\ldots,A_{N_A}\in\{\pm 1\}^{N_A}\\ B_1,\ldots,B_{N_B}\in\{\pm 1\}^{N_B}}}\naw{\sum_{i=1}^{N_A}A_i\vec{v}_i}\cdot\naw{\sum_{j=1}^{N_B}B_j\vec{w}_j}\label{14}
\end{equation}
where we have redefined $A_i\rightarrow-A_i$. Therefore, $C$ is expressed in terms of scalar products of vectors which are linear combinations, with the coefficients $\pm 1$, of the elements of Alice (Bob) orbit. The set of such vectors may be classified in group-theoretical terms. Let us consider a particular combination (for definiteness, we consider Alice orbit)
\begin{equation}
\sum_{i=1}^{N_A}A_i\vec{v}_i,\quad A_i=\pm 1.\label{15}
\end{equation}
Let $N_A^+$ ($N_A^-$) be the number of $+1$ ($-1$) coefficients entering (\ref{15}), $N_A^++N_A^-=N_A$. The group $G$ acts as some subgroup of permutations of the vectors $\vec{v}_i$. Therefore, by acting with $G$ on the combination (\ref{15}) one obtains the set of vectors of the same form with fixed $N_A^{\pm}$. One concludes that the set of vectors (\ref{15}) with $N_A^{\pm}$ fixed decomposes into disjoint sum of $G$ orbits. Accordingly, the set of all vectors of the above form is also a disjoint sum of such orbits. In order to determine their form it is sufficient to consider the configurations with $N_A^+ \leq N_A^-$; the remaining one are obtained by applying space inversion. Once the numbers $N_A^+\leq N_A^-$ are chosen one picks a particular combination (\ref{15}). In order to find the orbit obtained by acting with the elements of $G$ on this combination one has to determine the stability subgroup. The latter contains those elements of $G$ which permute $\vec{v}_i$'s with the same coefficients. However, in general it is larger; this is because the initial orbit $\poisson{\vec{v}_i}_{i=1}^{N_A}$ may contain the subset of vectors which are linearly dependent with the coefficients $\pm 1$ (this is, for example, the case for the cube which contains four pairs of opposite vectors). The same procedure may be applied to Bob's orbit. Once this is completed one can compute $C$ from eq.~(\ref{14}).
Contrary to the quantum bound the above described procedure does not seem to be more effective than direct evaluation of the bound $C$ from eq. (\ref{13}). However, it provides a nice geometrical picture of classical configurations. What is more it can serve for finding new orbits generated by the same group in order to use them as a starting point in our construction, i.e. as the initial orbits of Alice and Bob.
As a simple example let us take $G=S_4$ and the regular tetrahedron as its orbit. $S_4$ is the symmetry group of Platonic tetrahedron; simply, it permutes its four vertices $\vec{v}_i$, $i=1,2,3,4$. Since any three of them are linearly independent the stability subgroup of any vector (\ref{15}) consists exactly of all permutations exchanging vectors $\vec{v}_i$ entering (\ref{15}) with the same sign. Moreover, since our $G=S_4$ involves all permutations of vertices there is only one orbit corresponding to each $N_A^+$. Now, $N_A^+\leq N_A^-$ takes only three values, $N_A^+=0,\,1,\,2$. For $N_A^+=0$ the sum (\ref{15}) is obviously 0. For $N_A^+=1$ one finds tetrahedron twice as large as the initial one. It is also easy to find the orbit corresponding to $N_A^+=2$. To this end we connect the middle points of all edges obtaining thus regular octahedron and take the octahedron four times as large. The length of the vectors forming the vertices of the latter equals $\frac{4}{\sqrt{3}}$.
The orbits with $N_A^+>N_A^-$ are obtained by inversion. Concluding, the 16 combinations (\ref{15}) form two trivial orbits of $S_4$, two tetrahedrons in dual position and one octahedron.
\begin{figure}[!h]
\centering
\subfloat[two trivial orbits (red point) corresponding to $N_A^+=0,4$; the vertices of black tetrahedron represent Alice's settings]
{\includegraphics[width=0.35\textwidth]{Fig1a.pdf}}
\hspace{1cm}
\subfloat[the orbit (red octahedron) corresponding to $N_A^+=2$]{
\includegraphics[width=0.4\textwidth]{Fig1c.pdf}}
\\
\subfloat[two orbits (red and blue tetrahedrons) corresponding to $N_A^+=1,3$]{
\includegraphics[width=0.55\textwidth]{Fig1b.pdf}}
\caption{The "classical" orbits for tetrahedron.}
\end{figure}
Assume now that Bob's directions are determined by the vertices of regular tetrahedron which provides another orbit of $S_4$ (within the same representation). Note that the tetrahedrons are obtained by acting with the group elements on the vectors possessing stability subgroups of order 6. $S_4$ has four such subgroups, all in one conjugacy class. Taking into account that by inverting the initial vectors one obtains the tetrahedrons in dual position we conclude that there are only two tetrahedrons in dual position which form $S_4$ orbits of order 4. For dual tetrahedrons the sets of vectors (\ref{15}) coincide.
Assuming that both Alice and Bob settings form regular tetrahedrons generated by the same representation of $S_4$ we find that their "classical" configurations coincide. Therefore, we infer from eq.~(\ref{14}) that the classical bound (\ref{14}) is simply the square of the length of the largest vector which gives $\frac{16}{3}$. On the other hand eq.~(\ref{12}) yields also $\frac{16}{3}$ as the quantum bound. Thus there is no violation of Bell's inequality in this case.
\section{Some examples}
Let us consider some examples. In all cases considered below the group $G$ generating the orbit is either $S_4$ or $O_h$. The threedimensional irreducible representations of both groups are explicitly described in Appendix. The sets $\{\vec{v}_i\}_{i=1}^{N_A}$, $\{\vec{w}_j\}_{j=1}^{N_B}$ of Alice and Bob settings are the orbits of $G$ generated by the action of the representation matrices on some carefully selected initial vectors; the latter must be the eigenvectors, corresponding to the eigenvalue 1, of elements belonging to some subgroups of $G$ (stability subgroups). These initial vectors, together with explicit description of the corresponding orbits for all cases considered below, are given in the Appendix. We also give there the example of classical vectors, $\sum_{i=1}^{N_A}A_i\vec{v}_i$ and $\sum_{i=1}^{N_B}B_i\vec{w}_i$, saturating the classical bound (\ref{14}) for the configuration tetrahedron (Alice) - octahedron (Bob), as well as the orbits they generate according to the discussion of the previous section.
The orbits we are considering are:
\begin{itemize}
\item tetrahedron - generated by $S_4$
\item octahedron - generated by $S_4$ or $O_h$
\item cube - generated by $O_h$
\item cuboctahedron - generated by $S_4$ or $O_h$
\item truncated octahedron - generated by $S_4$ or $O_h$.
\end{itemize}
The algorithm described in the previous section can be directly applied to all pairs of the above solids except the pair tetrahedron - cube (see, however, below). The results are presented in Table \ref{t1}.
\begin{table}
\begin{tabular}{|l|c|c|}
\hline
Alice - Bob & Classical bound & Quantum value\\
\hline
cuboctahedron - tetrahedron & 13.0639 & 16\\
cuboctahedron - octahedron & 16.9706 & 24\\
cuboctahedron - cube & 26.1279 & 32\\
cuboctahedron - cuboctahedron & 40 & 48\\
truncated octahedron - tetrahedron & 24.7871 & 32\\
truncated octahedron - octahedron & 42.9325 & 48\\
truncated octahedron - cube & 49.5742 & 64\\
truncated octahedron - cubocthedron & 75.8947 & 96\\
truncated octahedron - truncated octahedron & 160 & 192\\
tetrahedron - octahedron & 6.9282 & 8\\
cube - octahedron & 13.8564 & 16\\
\hline
\end{tabular}
\caption{Classical bounds and quantum values for various combinations of Alice vs. Bob orbits.}\label{t1}
\end{table}
One can compare some of the above results with those obtained by Tavakoli and Gisin \cite{Tavakoli}. For example, for cube - octahedron we find the same value. On the other hand in the tetrahedron - octahedron case the classical bound obtained here is better than the one quoted in \cite{Tavakoli} - 7.82. This is because the relative positions of solids differ: in \cite{Tavakoli} they are chosen to be those given by "PolyhedronData" of Mathematica's software while here they are in position, described in previous section, resulting from group-theoretical considerations.
The "classical" numbers presented in Table \ref{t1} are approximate values. Indeed, from eqs.~(\ref{13}), (\ref{14}) and the forms of $\vec{v}_i$'s and $\vec{w}_j$'s presented in Appendix it follows that they are expressible in terms of square roots of integers. For example, for octahedron - cuboctahedron the ratio quantum - to classical bound is $1.4142\simeq\sqrt{2}$, the same as for CHSH inequality. It is likely that in this case the set of all Alice and Bob settings can be decomposed into the subsets leading to the original CHSH inequality (cf. Sec.~8 of Ref \cite{Tavakoli}). On Fig.~\ref{Fig2} we present some pairs of orbits.
\begin{figure}[!h]
\subfloat[cuboctahedron - tetrahedron]{\includegraphics[width=0.5\textwidth]{Fig2a.pdf}}
\subfloat[cuboctahedron - octahedron]{\includegraphics[width=0.5\textwidth]{Fig2b.pdf}}\\
\subfloat[truncated octahedron - tetrahedron]{\includegraphics[width=0.5\textwidth]{Fig2c.pdf}}
\subfloat[truncated octahedron - octahedron]{\includegraphics[width=0.5\textwidth]{Fig2d.pdf}}
\caption{Examples of Alice (black) and Bob (red) settings.}
\label{Fig2}
\end{figure}
Let us note that even the case tetrahedron - cube can be dealt with grouptheoretical method. The cube obtained by acting with elements of $O_h$ on the initial vector may be viewed as two tetrahedrons in dual position (see Fig. \ref{Fig3}).
Now, cube - cube configuration leads to $\frac{64}{3}$ both for classical and quantum cases. Obviously, for tetrahedron - cube configuration one finds one half of this result, i.e.~$\frac{32}{3}$.
\begin{figure}[!htp]
\centering\includegraphics[scale=0.5]{Fig3}
\caption{Cube as two (red and black) tetrahedrons in dual position.}\label{Fig3}
\end{figure}
Finally, let us come back to the case of two tetrahedrons. We found no violation of Bell inequality because tetrahedrons corresponding to Alice and Bob settings must be dual. The reason for that is that they are nongeneric orbits with stability groups of order 6. This fact strongly reduces the number of possibilities. Therefore, we should look for the group for which the generic orbit has four vertices i.e.~a group of order 4. Now, the regular tetrahedron is generated by cyclic subgroup of $S_4$ of order four, generated by cyclic permutation $(1234)$. So let us take $G=Z_4$ to be cyclic group of order four. Its threedimensional representation can be obtained from that of $S_4$ by subducing it to $Z_4$. In the suitably chosen basis it takes the form (for simplicity we write simply $g$ instead of $D(g)$):
\begin{equation}
\begin{split}
& e=\mathbbm{I},\quad g_1\equiv g=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & 0 & 1\\
0 & -1 & 0
\end{array}\right),\quad g_2\equiv g^2=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & -1
\end{array}\right), \\
& g_3\equiv g^3=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0
\end{array}\right).
\end{split}\label{16}
\end{equation}
This representation is reducible as the sum of one- and twodimensional ones. The generic orbits consist of four vertices. Taking
\begin{equation}
\vec{v}_1=\left(\begin{array}{c}
a\\
b\\
c
\end{array}\right ),\quad a^2+b^2+c^2=1\label{17}
\end{equation}
and denoting $\vec{v}_{k+1}=g_k\vec{v}_1$, $k=1,2,3$, one finds
\begin{equation}
\vec{v}_2=\left(\begin{array}{c}
-a\\
c\\
-b
\end{array}\right ),\quad \vec{v}_3=\left(\begin{array}{c}
a\\
-b\\
-c
\end{array}\right ), \quad \vec{v}_4=\left(\begin{array}{c}
-a\\
-c\\
b
\end{array}\right ).\label{18}
\end{equation}
Note that $\vec{v}_1\cdot\vec{v}_2=\vec{v}_1\cdot\vec{v}_4=\vec{v}_2\cdot\vec{v}_3=\vec{v}_3\cdot\vec{v}_4=-a^2\equiv\cos\psi$, $\vec{v}_1\cdot\vec{v}_3=\vec{v}_2\cdot\vec{v}_4=a^2-b^2-c^2=\cos\varphi$, $(\vec{v}_2-\vec{v}_4)\cdot(\vec{v}_1-\vec{v}_3)=0$. The resulting orbit is depicted on Fig.~\ref{Fig4}.
\begin{figure}
\begin{center}\includegraphics[width=0.6\textwidth]{Fig4a.pdf}\end{center}
\caption{The generic orbit of $Z_4$; $\measuredangle 1O3=\measuredangle 2O4=\varphi$, $\measuredangle 1O2=\measuredangle 1O4=\measuredangle 2O3=\measuredangle 3O4=\psi$.}
\label{Fig4}
\end{figure}
In particular, taking $a^2=b^2=c^2=\frac{1}{3}$ one obtains the regular tetrahedron.
Let us assume that the Alice orbit is the regular tetrahedron (say $a=b=c=\frac{1}{\sqrt{3}}$) while Bob's orbit corresponds to the vector $\vec{w}$ with arbitrary $a$, $b$, $c$ obeying $a^2+b^2+c^2=1$. First we determine the quantum bound. Eq.~(\ref{12}) cannot be applied directly since the representation is reducible. Actually, the orthogonality relations are valid for the representations irreducible in the complex domain. $Z_4$ is abelian so each representation irreducible in the complex domain is onedimensional. In fact $g_1=g$ (and, consequently, all $g_{{\alpha}}$, ${\alpha}=1,2,3$) can be diagonalized using the unitary matrix
\begin{equation}
U=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\
0 & \frac{i}{\sqrt{2}} & -\frac{i}{\sqrt{2}}
\end{array}\right )\label{19}
\end{equation}
so that ($\tilde{g}_\alpha\equiv U^+gU$, $\alpha=1,2,3$)
\begin{equation}
\tilde{g}\equiv U^+gU=\left(\begin{array}{ccc}
-1 & 0 & 0\\
0 & i & 0\\
0 & 0 & -i
\end{array}\right ).\label{20}
\end{equation}
Let $e_\mu(\tilde{g}_\alpha)$, $\mu=1,2,3$, be the character corresponding to the first, second and third row of $\tilde{g}_\alpha$, respectively. Then any matrix $\tilde{g}_\alpha$ can be written as
\begin{equation}
\naw{\tilde{g}_\alpha}_{ij}=\sum_{\mu=1}^3e_\mu(\tilde{g}_\alpha)\delta_{i\mu}\delta_{j\mu}. \label{21}
\end{equation}
Using eq.~(\ref{21}) and the orthogonality relations
\begin{equation}
\sum_{g_\alpha\in Z_4}\overline{e_\mu(\tilde{g}_\alpha)}e_\nu(\tilde{g}_\alpha)=4\delta_{\mu\nu}\label{22}
\end{equation}
one finds
\begin{equation}
\sum_{g_\alpha,g_\beta\in Z_4}\naw{g_\alpha\vec{v},g_\beta\vec{w}}^2=4\sum_{g_\alpha\in G}\naw{\vec{v},g_\alpha\vec{w}}^2=4\sum_{g_\alpha\in Z_4}\modu{\naw{\tilde{\vec{v}},\tilde{g}_\alpha\tilde{\vec{w}}}}^2=16\sum_{i=1}^3\modu{\tilde{v}_i}^2\modu{\tilde{w}_i}^2 \label{23}
\end{equation}
where $\tilde{\vec{v}}\equiv U^+\vec{v}$, $\tilde{\vec{w}}\equiv U^+\vec{w}$. Now, $v_i=\frac{1}{\sqrt{3}}$, $i=1,2,3$ and by virtue of eq.~(\ref{19}), $\modu{\tilde{v}_i}=\frac{1}{\sqrt{3}}$. Therefore, eq.~(\ref{23}) yields $\frac{16}{3}$ as the quantum bound. As far as the classical bound is concerned one may either classify all orbits consisting of the vectors of the form (\ref{15}) and use eq.~(\ref{14}) (again quite nice geometrical picture emerges) or use directly eq.~(\ref{13}).
The final result reads
\begin{equation}
C=\frac{1}{\sqrt{3}}\max\naw{16|a|,8(\modu{b}+\modu{c})}.\label{24}
\end{equation}
Minimal value of $C$ is acquainted, for example, for $b=\pm\frac{2}{\sqrt{5}}$, $c=0$; then
\begin{equation}
C=\frac{16}{\sqrt{15}}<\frac{16}{3}\label{25}
\end{equation}
and the Bell inequality is violated.
\section{Summary}
Tavakoli and Gisin presented a very nice picture relating various settings for Alice and Bob, leading to the violation of Bell inequalities, to the geometry of Platonic solids. We have shown here that one can take as a starting point the symmetry groups of Platonic solids or, more precisely, the groups generating these solids as orbits of their threedimensional real representations. This point of view has some advantages. First, it allows the generalization to various groups and generic and nongeneric orbits (including, for example, in unified way the Archimedean solids).
Second, it leads to extremally simple expression for the quantum value of $\mathcal{B}$, given by eq.~(\ref{5}), in terms of the number od Alice's and Bob's settings only (eq.~(\ref{12})). Third, it provides a nice picture of the configurations (eq.~(\ref{15})) entering the formula (\ref{14}) for the classical bound. It should be also noted that the grouptheoretical approach determines some natural relative positions of the solids determining Alice's and Bob's settings.
Finally, let us show that the choice of $\ket{\phi}$ instead of $\ket{\phi^+}$, used in Ref.~\cite{Tavakoli}, does not influence the results. $\ket{\phi^+}$ belongs to the triplet representation of the rotation group; consequently, the relevant correlation function is no longer rotationally invariant
\begin{equation}
\bra{\phi^+}(\vec{v}_i\cdot\vec{\sigma})\otimes(\vec{w}_j\cdot\vec{\sigma})\ket{\phi^+}=\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j\label{26}
\end{equation}
where $\mathcal{I}_y$ is the reflection in $x-z$ plane. Instead of $\mathcal{B}$ defined by eq.~(\ref{5}) we consider the following one
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j}\av{A_iB_j}.\label{27}
\end{equation}
In particular, for the $\ket{\phi^+}$ state we get
\begin{equation}
\mathcal{B}=\sum_{i=1}^{N_A}\sum_{j=1}^{N_B}\naw{\vec{v}_i\cdot\mathcal{I}_y\vec{w}_j}^2.\label{28}
\end{equation}
Now, repeating the previous reasoning we find
\begin{equation}
\begin{split}
&\sum_{g,g'\in G}\naw{D(g)\vec{v}\cdot\mathcal{I}_y D(g')\vec{w}}^2\\
&=\sum_{a,b,c,d,e,f=1}^3 v_b v_e w_c'w_f'\sum_{g\in G}D(g)_{ab}D(g)_{de}\sum_{g'\in G}\naw{\mathcal{I}_yD(g')\mathcal{I}_y}_{ac}\naw{\mathcal{I}_yD(g')\mathcal{I}_y}_{df}\\
& =\sum_{a,b,c,d,e,f=1}^3 v_bv_ew_c'w_f'\frac{|G|^2}{9}\delta_{ad}\delta_{be}\delta_{ad}\delta_{cf}=\frac{|G|^2}{3}
\end{split}\label{29}
\end{equation}
because $D(g')$ and $\mathcal{I}_yD(g')\mathcal{I}_y$ are equivalent representations and $\vec{w}'=\mathcal{I}_y\vec{w}$.
As far as classical bound is concerned both formulae, eq.~(\ref{5}) and (\ref{27}), give the same results if at least one of the orbits (i.e.~defining Alice's and/or Bob's settings) is invariant under reflection in $x-z$ plane. This is, for example, the case for all pairs of Platonic solids considered in \cite{Tavakoli} which lead to Bell inequality violation.
\begin{appendices}
\section{}
Below we present all solids which form the $G$ - orbits appearing in the text.
\begin{figure}[!h]
\subfloat[the tetrahedron]{\centering\includegraphics[width=0.33\textwidth]{tetrahedron.pdf}}
\subfloat[the octahedron]{\includegraphics[width=0.33\textwidth]{octahedron.pdf}}
\subfloat[the cube]{\includegraphics[width=0.33\textwidth]{cube.pdf}}\\
\centering
\subfloat[the cuboctahedron]{\includegraphics[width=0.4\textwidth]{cuboctahedron.pdf}}
\subfloat[the truncated octahedron]{\includegraphics[width=0.4\textwidth]{truncated_octahedron.pdf}}
\caption{Solids which form the $S_4$-orbits or $O_h$-orbits.}
\end{figure}
\begin{enumerate}
\item The tetrahedron:
\begin{itemize}
\item[] Symmetry group: $S_4$
\item[] Orbit generating group: $S_4$ (or $Z_4$)
\item[] The initial vector: $\vec{v}_1=(1,0,0)$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=(1,0,0), & v_2=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}}\\
v_3\naw{-\frac{1}{3},\frac{2\sqrt{2}}{3},0}, & v_4\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}}. \end{array}\end{displaymath}
\end{itemize}
\item The octahedron:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}},0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{6}},\frac{1}{\sqrt{2}}}, &
v_2=\naw{\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{2}}},\\
v_3=\naw{\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}},0}, & v_4=\naw{-\frac{1}{\sqrt{3}},-\sqrt{\frac{2}{3}},0},\\
v_5=\naw{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{2}}}, & v_6=\naw{\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{6}},\frac{1}{\sqrt{2}}}. \end{array}\end{displaymath}
\end{itemize}
\newpage
\item The cube:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $O_h$
\item[] The initial vector: $v_1=\naw{1,0,0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=(1,0,0), & v_2=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}},\\
v_3=\naw{-\frac{1}{3},\frac{2\sqrt{2}}{3},0}, & v_4=\naw{-\frac{1}{3},-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}},\\
v_5=(-1,0,0), & v_6=\naw{\frac{1}{3},\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}}},\\
v_7=\naw{\frac{1}{3},-\frac{2\sqrt{2}}{3},0}, & v_8=\naw{\frac{1}{3},\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}}}.\end{array}\end{displaymath}
\end{itemize}
\item The cuboctahedron:
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}},0}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{-\sqrt{\frac{2}{3}},\frac{1}{\sqrt{3}},0},& v_2=\naw{0,\frac{\sqrt{3}}{2},\frac{1}{2}},\\
v_3=\naw{0,0,1},& v_4=\naw{-\sqrt{\frac{2}{3}},-\frac{1}{2\sqrt{3}},\frac{1}{2}}, \\
v_5=\naw{\sqrt{\frac{2}{3}},-\frac{1}{\sqrt{3}},0},&
v_6=\naw{0,-\frac{\sqrt{3}}{2},-\frac{1}{2}},\\
v_7=\naw{0,0,-1}, &v_8=\naw{\sqrt{\frac{2}{3}},\frac{1}{2\sqrt{3}},-\frac{1}{2}},\\
v_9=\naw{0,-\frac{\sqrt{3}}{2},\frac{1}{2}}, &
v_{10}=\naw{0,\frac{\sqrt{3}}{2},-\frac{1}{2}},\\
v_{11}=\naw{\sqrt{\frac{2}{3}},\frac{1}{2 \sqrt{3}},\frac{1}{2}}, &
v_{12}=\naw{-\sqrt{\frac{2}{3}},-\frac{1}{2 \sqrt{3}},-\frac{1}{2}}.\end{array}\end{displaymath}
\end{itemize}
\item The truncated octahedron
\begin{itemize}
\item[] Symmetry group: $O_h$
\item[] Orbit generating group: $S_4$ or $O_h$
\item[] The initial vector: $v_1=\naw{\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}$
\item[] The vertices:
\begin{displaymath}\begin{array}{ll}
v_1=\naw{\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}, & v_2=\naw{\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, \\ v_3=\naw{\sqrt{\frac{3}{5}},0,-\sqrt{\frac{2}{5}}}, &
v_4=\naw{\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_5=\naw{\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, & v_6=\naw{\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_7=\naw{-\sqrt{\frac{3}{5}},0,\sqrt{\frac{2}{5}}}, & v_8=\naw{-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, \\
v_9=\naw{-\sqrt{\frac{3}{5}},0,-\sqrt{\frac{2}{5}}}, &
v_{10}=\naw{-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\ v_{11}=\naw{-\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},\frac{1}{\sqrt{10}}}, & v_{12}=\naw{-\sqrt{\frac{3}{5}},-\sqrt{\frac{3}{10}},-\frac{1}{\sqrt{10}}},\\
v_{13}=\naw{-\frac{1}{\sqrt{15}},2 \sqrt{\frac{2}{15}},-\sqrt{\frac{2}{5}}},&
v_{14}=\naw{\frac{1}{\sqrt{15}},\sqrt{\frac{5}{6}},-\frac{1}{\sqrt{10}}},\\
v_{15}=\naw{-\frac{1}{\sqrt{15}},2 \sqrt{\frac{2}{15}},\sqrt{\frac{2}{5}}}, & v_{16}=\naw{\frac{1}{\sqrt{15}},\sqrt{\frac{5}{6}},\frac{1}{\sqrt{10}}},\\
v_{17}=\naw{\frac{1}{\sqrt{15}},-2 \sqrt{\frac{2}{15}},\sqrt{\frac{2}{5}}}, & v_{18}=\naw{-\frac{1}{\sqrt{15}},-\sqrt{\frac{5}{6}},\frac{1}{\sqrt{10}}},\\
v_{19}=\naw{\frac{1}{\sqrt{15}},-2 \sqrt{\frac{2}{15}},-\sqrt{\frac{2}{5}}}, & v_{20}=\naw{-\frac{1}{\sqrt{15}},-\sqrt{\frac{5}{6}},-\frac{1}{\sqrt{10}}},\\
v_{21}=\naw{-\frac{1}{\sqrt{15}},\frac{1}{\sqrt{30}},\frac{3}{\sqrt{10}}}, & v_{22}=\naw{\frac{1}{\sqrt{15}},-\frac{1}{\sqrt{30}},\frac{3}{\sqrt{10}}},\\
v_{23}=\naw{\frac{1}{\sqrt{15}},-\frac{1}{\sqrt{30}},-\frac{3}{\sqrt{10}}}, & v_{24}=\naw{-\frac{1}{\sqrt{15}},\frac{1}{\sqrt{30}},-\frac{3}{\sqrt{10}}}.\end{array}\end{displaymath}
\end{itemize}
\end{enumerate}
Below we give an example of "classical" vectors (\ref{15}) saturating the bound (\ref{14}) for the settings: tetrahedron (Alice) and octahedron (Bob). The $G$-orbits they generate are also presented.\\
Alice's vector: $\vec{V}\equiv\sum\limits_{i=1}^{4}A_i\vec{v}_i=\vec{v}_1-\vec{v}_2+\vec{v}_3+\vec{v}_4=\naw{\frac{2}{3},-\frac{4\sqrt{2}}{3},0}$\\
Bob's vector: $\vec{W}\equiv\sum\limits_{j=1}^{6}B_j\vec{w}_j=-\vec{w}_1+\vec{w}_2-\vec{w}_3+\vec{w}_4-\vec{w}_5+\vec{w}_6=\naw{\frac{2}{\sqrt{3}},-4\sqrt{\frac{2}{3}},0}$\\
The $S_4$ orbits generated by Alice's and Bob's vectors are depicted on Fig. \ref{Fig10}.
\begin{figure}[!h]
\subfloat[the initial Alice's settings (magenta) and the $S_4$ orbit generated by the vector $\vec{V}$]{\includegraphics[width=0.5\textwidth]{AppenRys1.pdf}}
\hspace{1 cm}\subfloat[the initial Bob's settings (black) and two $S_4$ orbits generated by the vectors $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys2.pdf}}\\
\subfloat[the $S_4$ orbits generated by $\vec{V}$, $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys3.pdf}}\hspace{1cm}
\subfloat[the initial Alice's and Bob's settings and the $S_4$ orbits generated by $\vec{V}$, $\vec{W}$ and $-\vec{W}$]{\includegraphics[width=0.5\textwidth]{AppenRys4.pdf}}
\caption{The $S_4$ orbits generated by Alice's (magenta and blue) and Bob's (black and red) vectors. Figs.~10c and 10d present the relative position of the Alice and Bob orbits generated by "classical" vectors.}\label{Fig10}\end{figure}
\section{}
In order to describe the threedimensional irreducible representation of $S_4$ group it is sufficient to write out the matrices representing transpositions. They read:
\begin{equation}
\begin{split}
& D\naw{12}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1
\end{array}\right],\qquad
D\naw{13}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\
0 & -\frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]\\
& D\naw{14}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{6}}{3}\\
-\frac{\sqrt{2}}{3} & \frac{5}{6} & -\frac{\sqrt{3}}{6}\\
-\frac{\sqrt{6}}{3} & -\frac{\sqrt{3}}{6}& \frac{1}{2}
\end{array}\right],\qquad
D\naw{23}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -\frac{1}{2} & \frac{\sqrt{3}}{2}\\
0 & \frac{\sqrt{3}}{2} & \frac{1}{2}
\end{array}\right]
\\
&D\naw{24}=\left[\begin{array}{ccc}
-\frac{1}{3} & -\frac{\sqrt{2}}{3} & \frac{\sqrt{6}}{3}\\
-\frac{\sqrt{2}}{3} & \frac{5}{6} & \frac{\sqrt{3}}{6}\\
\frac{\sqrt{6}}{3} & \frac{\sqrt{3}}{6}& \frac{1}{2}
\end{array}\right],\qquad
D\naw{34}=\left[\begin{array}{ccc}
-\frac{1}{3} & \frac{\sqrt{8}}{3} & 0\\
\frac{\sqrt{8}}{3} & \frac{1}{3} & 0\\
0 & 0 & 1
\end{array}\right].\end{split}\label{30}
\end{equation}
The $O_h$ group it the direct product of $S_4\times S_2$; for this reason the threedimensional irreducible representation of this group can be obtained by multiplying all 24 matrices representing $S_4$ by $\pm 1$.
\end{appendices}
\newpage
| train/arxiv |
BkiUaF825V5hcGj03HSJ | 5 | 1 | \section{Introduction}
\label{sec:intro}
Research on wave energy converters (WECs) has concentrated traditionally on systems of small floating bodies, like offshore heaving buoys (see \cite{EV76}--\cite{ FA02}). However, the seminal theories on WECs that originated from this first scientific approach to wave energy extraction in the 1970s, do not capture exhaustively the dynamics of the last-generation WECs. The latter are usually large-scale devices designed to be deployed in arrays, some of them in the near-shore environment. For example, while studying the dynamics of an offshore heaving WEC in a channel, Srokosz (1980) \cite{SR80} showed that resonance of the channel sloshing modes is detrimental to the efficiency of power absorption. Conversely, in a recent analysis of a large flap-type WEC in a channel, Renzi \& Dias \cite{RD12} noted that the trapping of transverse modes near the flap increases the efficiency of the converter. Because of the image effect of the channel walls, this fact is also expected to occur in an infinite array of flap-type converters. The aim of this work is to discover the dynamics of a system of last-generation flap-type WECs and to outline its similarities and differences with respect to the systems of the first generation. As a result of this analysis, an optimisation criterion for an array of flap-type WECs is devised, depending on the physical and geometrical parameters of the system.
In Section \ref{sec:model} the behaviour of an array of converters in the open ocean is investigated by taking as a reference the theoretical framework of Renzi and Dias \cite{RD12}. The expressions of the free-surface elevation for the diffracted and radiated wave field in the fluid domain are derived accordingly. Analysis of the wave motion in the far field allows to obtain new formulae for the reflection, transmission and radiation coefficients. Various relations between the hydrodynamic coefficients are then shown in Section \ref{sec:rel}. Some of these relations correspond directly to Srokosz's results \cite{SR80} for floating bodies of symmetric shape in a channel. Some others, on the other hand, incorporate specific properties of the wave field generated by the flap-type converter, not considered before, and point out the peculiarity of such WEC with respect to the converters of the first generation. The analytical model is validated against known theories in the small-gap and in the point-absorber limit. In Section \ref{sec:paran}, a parametric analysis is undertaken for optimising the performance of the system. It is shown that the maximum capture factor is attained at complete trapping of the transverse modes of the array. When complete trapping is not possible, partial trapping can still increase the performance of the system. Finally, in Section \ref{sec:waven} a practical application of an array of large flap-type WECs is devised. Comparison with available data obtained with a finite-element numerical code is very satisfactory (see \ref{sec:appD}).
\section{Mathematical model}
\label{sec:model}
\subsection{Theoretical background}
Consider an in-line array of identical flap-type wave energy converters, each hinged on a bottom foundation of height $c'$ in an ocean of constant depth $h'$, as shown in figure \ref{fig:geom}.
\begin{figure}
\centerline{\includegraphics[width=9cm, trim=3.5cm 0cm 3.5cm 0cm]{geometry.eps}}
\caption{Geometry of the array ($a$) and the reference flap ($b$) in physical variables.}
\label{fig:geom}
\end{figure}
Primes denote dimensional quantities. Monochromatic incident waves of amplitude $A_I'$, period $T'$ and frequency $\omega'=2\pi/T'$ are incoming from the right and set the flaps into motion, which is converted into useful energy by means of generators linked to each device. Since the practical applications of such a system are usually in the nearshore \citep{WF12}, where wave fronts are almost parallel to the shoreline because of refraction, normal incidence is also assumed. Let $w'$ and $b'$ be the width of each flap and the spatial period of the array, respectively. Then the gap between two consecutive flaps is $a'=b'-w'$ (see again figure \ref{fig:geom}). A Cartesian coordinate system is set, with the $x'$ direction orthogonal to the flaps, the $y'$ axis along the flap lineup and the $z'$ axis rising from the undisturbed water level $z'=0$, positive upwards; $t'$ denotes time. Due to periodicity, the origin of the system can be set arbitrarily on any flap, which is therefore identified as the reference flap. The analysis is performed in the framework of a linear inviscid potential-flow theory for small-amplitude oscillations. The velocity potential $\Phi'$ must satisfy the Laplace equation
\begin{equation}
\nabla'^2\Phi'(x',y',z',t') =0
\label{eq:lapl}
\end{equation}
in the fluid domain. The linearised kinematic-dynamic boundary condition on the free surface reads
\begin{equation}
\Phi'_{,t't'}+g \Phi'_{,z'}=0, \quad z'=0,
\label{eq:bcsurf}
\end{equation}
where $g$ is the acceleration due to gravity and subscripts with commas denote differentiation with respect to the relevant variables. Absence of normal flux at the bottom yields
\begin{equation}
\Phi'_{,z'}=0, \quad z'=-h'.
\label{eq:bottom}
\end{equation}
Because of normal incidence of the incoming wave field, periodicity of the problem requires
\begin{equation}
f'(x',y'+mb',z',t')=f'(x',y',z',t'),\quad m=0,\pm 1,\pm 2,\dots,\; y'\in(-b'/2,b'/2),
\label{eq:period}
\end{equation}
where $f'$ indicates any physical quantity associated to the problem and $m$ each of the flaps; $m=0$ denotes the reference flap. Extension to an obliquely-incident wave field can be easily made \citep[see for example][]{PE96}. However, since flap-type WECs are usually designed to operate under normally-incident waves \citep[see][]{WF12}, only normal incidence will be considered here. Because of the periodicity condition (\ref{eq:period}), the solution to the complete problem can be obtained by investigating the wave interaction with the reference flap centred at the origin, with $|y'|<b'/2$. Symmetry of the problem requires also
\begin{equation}
\Phi'_{,y'}=0,\quad y'=\pm b'/2,
\label{eq:simm}
\end{equation}
which can be regarded as a no-flux boundary condition on two imaginary waveguides at $y'=\pm b'/2$ (see again figure \ref{fig:geom}). Let $\theta'(t')$ be the angle of rotation of the flap, positive if anticlockwise; then the kinematic boundary condition on the flap yields
\begin{equation}
\Phi'_{,x'}=-\theta'_{,t'}(t') (z'+h'-c') H(z'+h'-c'), \quad x'=\pm 0, |y'|<w'/2,
\label{eq:plate}
\end{equation}
where the thin-body approximation has been applied \citep{LM01}. The Heaviside step function in (\ref{eq:plate}) assures absence of flux through the bottom foundation. The problem defined above is formally equivalent to that solved by Renzi and Dias \cite{RD12} for a single converter in a channel. Here the main arguments of the theory in \cite{RD12} are retraced and applied to the array configuration. First, the system (\ref{eq:lapl})--(\ref{eq:plate}) is non-dimensionalised as follows \citep[see][eqn. (2.1)]{RD12}
\begin{eqnarray}
&(x,y,z,h,w,a,c)=(x',y',z',h',w',a',c')/b',\; t=\sqrt{g/b'}\,t',\nonumber \\
&\Phi=\left(\sqrt{gb'}A'\right)^{-1}\Phi',\; \theta=(b'/A') \theta',
\label{eq:nondimvar}
\end{eqnarray}
where the wave amplitude scale $A'\ll b'$ because of the hypothesis of small-amplitude oscillations. In expression (\ref{eq:nondimvar}), $a=(1-w)\in(0,1)$ defines the aperture of the array. Then time is factored out by setting
\begin{equation}
\Phi(x,y,z,t)=\Re\left\lbrace \phi(x,y,z) e^{-i\omega t}\right\rbrace,\quad \theta(t)=\Re\left\lbrace \Theta e^{-i\omega t}\right\rbrace,
\label{eq:separ}
\end{equation}
with $\omega=\sqrt{b'/g}\,\omega'$ \citep[see][eqn. (2.11)]{RD12}. The global spatial potential
$$
\phi=\phi^R+\phi^S
$$
is the sum of the radiation potential $\phi^R$ and the scattering potential $\phi^S$. The latter is in turn decomposed into
$$\phi^S=\phi^I+\phi^D,$$ where
\begin{equation}
\phi^I(x,y,z)=-\frac{i A_I}{\omega\cosh kh}\cosh k(z+h) e^{-i kx}
\label{eq:incidentwav}
\end{equation}
is the incident wave potential and $\phi^D$ is the diffraction potential. $\phi^R$ and $\phi^D$ must be both outgoing at large $|x|$. In (\ref{eq:incidentwav}), $A_I=A'_I/A'$ is the non-dimensional amplitude of the incident wave and $k$ is the wavenumber, corresponding to the real solution of the dispersion relationship $\omega^2=k \tanh kh$.
Following the method described in Appendix B of \cite{RD12}, application of the Green integral theorem yields two hypersingular integral equations, in terms of the jump in radiation and scattering potentials across the plate \citep[see][eqn. (B10)]{RD12}. Those equations are solved by expanding the jumps in potential into series of Chebyshev polynomials of even order \citep[for details, see eqn.s (B11)--(B18) of][]{RD12}. Careful treatment of the singularity \citep[see][eqn. (B19)]{RD12} ultimately allows to write the potentials in the reference domain $|y|<1/2$ in a new semi-analytical form \citep[see][eqn.s (B24) and (B25)]{RD12}. The radiation potential is
\begin{equation}
\phi^R(x,y,z)=\sum_{n=0}^{+\infty}\sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^R_{npm}(x,y,z),
\label{eq:radpotsum}
\end{equation}
where
\begin{eqnarray}
\phi^R_{npm}(x,y,z)&=&-\frac{i w V}{8}\,\kappa_n x Z_n(z) \alpha_{(2p)n}\int_{-1}^{1} \left(1-u^2\right)^{1/2} \nonumber\\
&\times& U_{2p}(u)
\frac{H_1^{(1)}\left(\kappa_n\sqrt{x^2+(y-\frac{1}{2} wu-m)^2} \right)}{\sqrt{x^2+(y-\frac{1}{2} wu-m)^2}}\,du,
\label{eq:radpot}
\end{eqnarray}
$H_1^{(1)}$ being the Hankel function of the first kind and first order. In (\ref{eq:radpot}), $V=i\omega\Theta$ is the complex angular velocity of the flap, the subscript $m$ identifies the contribution of each single flap, while the subscript $p$ indicates the order of the Chebyshev expansion, $U_{2p}$ being the Chebyshev polynomial of the second kind and even order $2p$, $p=0,1,...,P\in\mathbb{N}$. The subscript $n$ identifies the contribution of each depth mode
\begin{equation}
Z_n(z)=\frac{\sqrt{2}\cosh \kappa_n (z+h)}{\left(h+\omega^{-2}\sinh^2 \kappa_n h \right)^{1/2}},\quad n=0,1,\dots,
\label{eq:Zn}
\end{equation}
where $\kappa_0=k$, while $\kappa_n=ik_n$ denote the complex solutions of the dispersion relationship
\begin{equation}
\omega^2=-k_n\tan k_nh,\quad n=1,2,\dots
\label{eq:disprel}
\end{equation}
Finally, the $\alpha_{(2p)n}$ are the complex solutions of a system of linear equations ensuring that $\phi^R$ satisfies the kinematic condition on the flap \citep[see][eqn.s (B22) and (B23)]{RD12}. This system is solved numerically with a collocation scheme, therefore the solution (\ref{eq:radpot}) is partly numerical. In summary, $\phi_{npm}$ (\ref{eq:radpot}) indicates the $n$-th depth mode, $p$-th order potential of the wave field radiated by the $m$-th flap ($m$-th array mode), moving at unison with all the other flaps. The diffraction potential is given by
\begin{equation}
\phi^D(x,y,z)=\sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^D_{pm},
\label{eq:difpotsum}
\end{equation}
where
\begin{eqnarray}
\phi^D_{pm}(x,y,z)&=&-\frac{i w A_I}{8}\,kx\, Z_0(z)\beta_{2p}\int_{-1}^{1} \left(1-u^2\right)^{1/2} \nonumber\\
&\times& U_{2p}(u)
\frac{H_1^{(1)}\left(k\sqrt{x^2+(y-\frac{1}{2} wu-m)^2} \right)}{\sqrt{x^2+(y-\frac{1}{2} wu-m)^2}}\,du.
\label{eq:difpot}
\end{eqnarray}
In the latter, the $\beta_{2p}$ are the complex solutions of a system of linear equations, which ensures that $\phi^D$ satisfies the no-flux condition on the flap \citep[see][eqn.s (B22) and (B23)]{RD12}. Again, $\phi^D_{pm}$ indicates the $p$-th order potential diffracted by the $m$-th flap, in the presence of all the other flaps. Note that in $\phi^D$ (\ref{eq:difpot}) only the $0$-th depth mode is present, as required by the solvability of the whole radiation-diffraction problem \citep[see Appendix B.2 of][]{RD12}. Computational aspects involved in the numerical evaluation of (\ref{eq:radpot}) and (\ref{eq:difpot}) are detailed in \S 2.2 of \cite{RD12}.
\subsection{Body motion}
The equation of motion of the reference flap in the frequency domain is that of a damped harmonic oscillator \citep[see eqn. (2.33) of][]{RD12}, namely
\begin{equation}
\left[-\omega^2(I+\mu)+C-i\omega(\nu+\nu_{pto})\right]\Theta=F,
\label{eq:bodymot}
\end{equation}
depending on the moment of inertia of the flap $I=I'/(\rho b'^5)$, on the flap buoyancy torque $C=C'/(\rho g b'^4)$ and on the power take-off (PTO) coefficient $\nu_{pto}=\nu'_{pto}/(\rho b'^4\sqrt{gb'})$, where $\rho$ is the water density. The latter parameters are assumed to be all known. In (\ref{eq:bodymot})
\begin{equation}
\mu=\frac{\pi w}{2\sqrt{2}}\,\Re\left\lbrace \sum_{n=0}^{\infty} \alpha_{0n} \frac{\kappa_n(h-c)\sinh \kappa_n h+\cosh \kappa_n c-\cosh\kappa_n h}{\kappa_n^2\left(h+\omega^{-2}\sinh^2\kappa_n h \right)^{1/2}} \right\rbrace
\label{eq:mu}
\end{equation}
is the added inertia torque \citep[see eqn. (2.34) of][]{RD12}, while
\begin{equation}
\nu=\frac{\pi w}{2\sqrt{2}}\Im\left\lbrace \alpha_{00} \right\rbrace \frac{\omega\left[k(h-c)\sinh k h+\cosh k c-\cosh k h\right]}{k^2\left(h+\omega^{-2}\sinh^2 kh \right)^{1/2}}
\label{eq:nu}
\end{equation}
and
\begin{equation}
F=-\frac{i\pi w A_I}{2\sqrt{2}}\beta_{0}\frac{\omega\left[k(h-c)\sinh kh+\cosh kc-\cosh kh\right]}{k^2\left(h+\omega^{-2}\sinh^2 kh\right)^{1/2}}
\label{eq:F}
\end{equation}
denote, respectively, the radiation damping \citep[see eqn. (2.35) of][]{RD12} and the complex exciting torque \citep[see eqn. (2.36) of][]{RD12}. If the PTO system is designed such that
$$\nu_{pto}=\sqrt{\frac{\left[C-(I+\mu)\omega^2\right]^2}{\omega^2}+\nu^2},$$
which corresponds to the optimum PTO damping \citep[see eqn. (2.40) of][]{RD12},
then the average generated power over a period is
\begin{equation}
P=\frac{1}{4}|F|^2\left[\sqrt{\frac{\left[C-(I+\mu)\omega^2\right]^2}{\omega^2}+\nu^2}+\nu\right]^{-1}.
\label{eq:pow}
\end{equation}
Now, the generated power (\ref{eq:pow}) is maximum under resonant amplification of the body motion, which occurs when
\begin{equation}
\omega=\sqrt{\frac{C}{I+\mu}}.
\label{eq:bodyres}
\end{equation}
By substitution of the latter expression into (\ref{eq:pow}), the optimum power available for extraction from each flap is therefore
\begin{equation}
P_{opt}=\frac{1}{8}\frac{|F|^2}{\nu},
\label{eq:Popt}
\end{equation}
which matches the well-known result of Srokosz \cite{SR80}.
The performance of each element of the array is assessed quantitatively by using two main factors. The amplitude factor
\begin{equation}
A_F=\frac{(h-c)\tan|\Theta|}{A_I}
\label{eq:AF}
\end{equation}
is defined as the ratio between the flap horizontal stroke and the amplitude of the incident waves, $\Theta$ being the solution of the equation of motion (\ref{eq:bodymot}). Finally, the capture factor is defined as the ratio between the power extracted per unit flap width and the power available per unit crest length
\begin{equation}
C_F=\frac{P}{\frac{1}{2} A_I^2 C_g w},
\label{eq:CF}
\end{equation}
where
\begin{equation}
C_g=\frac{\omega}{2k}\left(1+\frac{2kh}{\sinh 2kh}\right)
\end{equation}
is the group velocity of the incident waves. Since $P=P(a)$ and $w=1-a$, the capture factor (\ref{eq:CF}) depends intrinsically on the aperture of the array. A strength point of the method of \cite{RD12} is that knowing the coefficients $\alpha_{0n}$ and $\beta_0$ is sufficient to obtain immediately all the physical quantities describing the performance of the device (eqn.s \ref{eq:mu}--\ref{eq:CF}), without need to evaluate the potentials (\ref{eq:radpotsum}) and (\ref{eq:difpotsum}).
The wave motion at large distance from the array will be now analysed.
\section{The far field}
In this section the behaviour of the wave field is investigated at large distance from the array. First, consider the radiation potential $\phi^R_{npm}$ given by (\ref{eq:radpot}). For $n>0$ the Hankel function in (\ref{eq:radpot}) can be rewritten as
\begin{eqnarray}
H_1^{(1)}\left(\kappa_n|x|\sqrt{1+\left(\frac{y}{x}-\frac{wu}{2x}-\frac{m}{x}\right)^2}\right)\nonumber\\
=-\frac{2}{\pi}K_1\left(\kappa_n|x|\sqrt{1+\left(\frac{y}{x}-\frac{wu}{2x}-\frac{m}{x}\right)^2} \right),
\label{eq:modbes}
\end{eqnarray}
where $K_n$ denotes the modified Bessel function of the second kind and order $n$ \citep[see \S 8.407 of][]{GR07}. Since $K_1(z)\propto e^{-z}$ as $z\rightarrow\infty$ in (\ref{eq:modbes}) and hence in (\ref{eq:radpot}), the argument of $\phi^R_{npm}$ for $n>0$ decays exponentially in the far field, so that at leading order
\begin{equation}
\phi^R_{npm}\sim 0, \quad |x|\rightarrow\infty\,,n>0.
\end{equation}
This happens since the modes $n>0$ physically represent the parasite waves generated by the motion of the flaps. These remain trapped near the device and do not contribute to the wave motion in the far field \cite{ME05}. As a consequence,
\begin{equation}
\phi^R\sim \sum_{p=0}^P\sum_{m=-\infty}^{+\infty}\phi^R_{0pm},\quad |x|\rightarrow\infty.
\label{eq:phirapp}
\end{equation}
Now substituting (\ref{eq:radpot}) into (\ref{eq:phirapp}), using the asymptotic expression (\ref{eq:Sff}) with $(X,Y)=(x,y-wu/2)$ and the integral formulae (\ref{eq:Ip0}), (\ref{eq:Ipq2}) and finally developing some straightforward algebra, yields
\begin{equation}
\phi^R\sim -\frac{iV}{\omega}\frac{\cosh k(z+h)}{\cosh kh}\,\sum_{q=0}^{\bar{q}}\mathcal{A}_q^{\pm}e^{\pm i\gamma_q kx} \cos(2q\pi y),\quad x\rightarrow\pm\infty.
\label{eq:phirff}
\end{equation}
In the latter expression, $\gamma_q=\sqrt{1-(2q\pi/k)^2}$ and $\bar{q}$ is the largest integer for which $\gamma_q$ is real, while
\begin{eqnarray}
\mathcal{A}_0^{\pm}&=&\mp\frac{i\pi}{8}\,w\,\omega\alpha_{00}Z_0(0),\label{eq:A0}\\
\mathcal{A}_q^{\pm}&=&\mp \frac{i}{4}\,w\,\omega Z_0(0)\, \epsilon_q\sum_{p=0}^{P} \alpha_{(2p)0}(-1)^p(2p+1)\frac{J_{2p+1}(q\pi w)}{qw}.\label{eq:Aq}
\end{eqnarray}
In (\ref{eq:Aq}), $\epsilon_q$ is the Jacobi symbol, while $J_{2p+1}$ is the Bessel function of first kind and order $2p+1$. Note that the radiation potential in the far field (\ref{eq:phirff}) is the sum of a progressive long-crested wave (term $q=0$) and several progressive short-crested waves (terms $0<q<\bar{q}$), which correspond to the propagating sloshing modes of the equivalent channel configuration of \cite{RD12}. Expression (\ref{eq:phirff}) is similar in form to (2.24) of \citep{SR80} (accounting for the various differences in the nomenclature), which gives the far-field expression of the radiation potential for a floating body, symmetric with respect to the $x$ axis, in a channel. In (2.24) of \cite{SR80}, however, the coefficients $\mathcal{A}_{q}^{\pm}$ are left in a general form, while here they are determined explicitly for the flap-type converter. The same steps can be repeated to find the far-field expression of the diffraction potential $\phi^D$ (\ref{eq:difpotsum}). By substituting (\ref{eq:difpot}), (\ref{eq:Sff}), (\ref{eq:Ip0}) and (\ref{eq:Ipq2}) in (\ref{eq:difpotsum}) and developing the algebra, the diffraction potential in the far field becomes
\begin{equation}
\phi^D(x,y)\sim \mp\frac{iA_I}{\omega}\frac{\cosh k(z+h)}{\cosh kh}\sum_{q=0}^{\bar{q}}R_q e^{\pm i\gamma_q kx}\cos(2q\pi y),\quad x\rightarrow\pm\infty,
\label{eq:phidff}
\end{equation}
where
\begin{eqnarray}
R_0&=&-\frac{i\pi}{8}\,w\,\omega\beta_0Z_0(0),\label{eq:R0}\\
R_q&=&-\frac{i}{4}\,w\,\omega Z_0(0)\,\epsilon_q\sum_{p=0}^P\beta_{2p}(-1)^p(2p+1)\frac{J_{2p+1}(q\pi w)}{qw}.\label{eq:Rq}
\end{eqnarray}
Equation (\ref{eq:phidff}) is similar to (2.25) of \cite{SR80}, in which, however, the $R_q$ are left in a general form.
Note that the calculation of the coefficients $\mathcal{A}_q^\pm$ and $R_q$ is straightforward once the linear system for $\alpha_{(2p)0}$ and $\beta_{2p}$ is solved \citep[see eqn. B23 of][]{RD12}.
\subsection{The free-surface elevation}
The amplitude of the free surface in the far field is an important parameter in order to assess the impact of the array on the wave climate of the surrounding area. Given the total potential $\Phi(x,y,t)$, the free-surface elevation is
$$\zeta(x,y,t)=-\Phi_{,t}|_{z=0}=\Re\left\lbrace\eta(x,y) e^{-i\omega t}\right\rbrace,$$
where
\begin{equation}
\eta(x,y)=i\omega(\phi^I+\phi^R+\phi^D)
\end{equation}
is the relevant complex spatial component. Substituting (\ref{eq:incidentwav}) for $\phi^I$ and the far-field expressions (\ref{eq:phirff}) and (\ref{eq:phidff}) for $\phi^R$ and $\phi^D$, respectively, yields
\begin{equation}
\eta(x,y)\sim \left\lbrace
\begin{tabular}{l l}
$A_Ie^{-ikx}+\sum_{q=0}^{\bar{q}}(A_I R_q+V\mathcal{A}_q^+)e^{i\gamma_q kx}\cos(2q\pi y)$, & $x\rightarrow+\infty$\\
$\sum_{q=0}^{\bar{q}}(A_I T_q+V\mathcal{A}_q^-)e^{-i\gamma_q kx}\cos(2q\pi y)$,& $x\rightarrow-\infty$
\end{tabular}
\right.,
\label{eq:etaff}
\end{equation}
where
\begin{equation}
T_0=1-R_0, \quad T_q=-R_q.
\label{eq:T0}
\end{equation}
Overall, the free-surface elevation is the sum of a long-crested wave (term $q=0$) and several short-crested waves (terms $0<q<\bar{q}$), namely the propagating transverse modes of the array. Physically, in (\ref{eq:etaff}) the terms $\mathcal{A}_q^{\pm}$ represent the $q$th-mode radiation coefficients, $R_q$ is the $q$th-mode reflection coefficient and finally $T_q$ represents the $q$th-mode transmission coefficient. They enjoy all the general properties of the analogous terms introduced by Srokosz \cite{SR80} for bodies of symmetric shape in a channel. In addition, such coefficients have some specific properties, peculiar to flap-type bodies, which derive from their analytical structure, as shown in detail in \S\ref{sec:rel}. Figure \ref{fig:porevcom} shows the behaviour of $R_0$ and $T_0$ against the nondimensional wavenumber $k$ for a typical configuration where $a=1/2$. The plots in figure \ref{fig:porevcom} compare favourably with those of Williams \& Crull \cite[fig. 3]{WC93} and Porter \& Evans \cite[fig. 2]{PE96}, who studied the scattering of incident waves by an array of thin screens.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm, trim= 5cm 9cm 5cm 0cm]{porevcom.eps}
\end{center}
\caption{Magnitude of the fundamental reflection and transmission coefficients, respectively $R_0$ (\ref{eq:R0}) and $T_0$ (\ref{eq:T0}), versus the non-dimensional wavenumber $k$. In this layout the flap width equals the gap size, i.e. $a=w=1/2$.
}
\label{fig:porevcom}
\end{figure}
Note the spiky behaviour of the coefficients, with spikes occurring at the resonant wavenumbers $k=2(\bar{q}+1) \pi$, $\bar{q}=0,1,\dots$ of the transverse short-crested waves, for which the $(\bar{q}+1)$th transverse mode turns from trapped to propagating.
In the following, the coefficients $R_0$, $T_0$ and $\mathcal{A}_0^\pm$ will be shown to enjoy some interesting properties and to be very useful for determining some relations between the hydrodynamic coefficients of the system.
\section{Derivation of relations for an array of flap-type WECs}
\label{sec:rel}
In this section, relations are derived for an array of flap-type WECs, based on the results obtained in the previous section. Some of these relations correspond directly to Srokosz's results \cite{SR80} for floating bodies of symmetric shape, while some others incorporate specific properties of the wave field (\ref{eq:etaff}) generated by the array of flap-type converters. In this sense, such expressions are new and point out the peculiarity of flap-type WECs with respect to the converters of the first generation.
\subsection{Extended Bessho-Newman relation}
First consider $\mathcal{A}_0^+$ and $R_0$. From (\ref{eq:A0}) and (\ref{eq:R0}) it is immediate to get, respectively,
$\arg(\mathcal{A}_0^+)=\arg(\alpha_{00})-\pi/2$ and $\arg(R_0)=\arg(\beta_0)-\pi/2$. Since $\arg(\alpha_{00})=\arg(\beta_0)$ \citep[see Appendix C of][]{RD12}, then the complex coefficients $\mathcal{A}_0^+$ and $R_0$ must have the same argument, say $\delta$, for which
\begin{equation}
\mathcal{A}_0^+=|\mathcal{A}_0^+|e^{i\delta},\quad R_0=|R_0|e^{i\delta},
\label{eq:delta}
\end{equation}
for any wavenumber $k$. The same relation, but only for $k<2\pi$, can be also derived from the Bessho-Newman relation
\begin{equation}
A_0^+-\sum_{q=0}^{\bar{q}}\frac{\gamma_q}{\epsilon_q}\left(A_q^{+\ast}R_q+A_q^{-\ast}T_q \right)=0,
\label{eq:BN}
\end{equation}
where $()^\ast$ denotes the complex conjugate. Expression (\ref{eq:BN}) is obtained by applying Green's integral theorem to $\phi^S$ and $(\phi^R-\phi^{R\ast})$ and corresponds to (3.2) of \citep{SR80}, with small variations due to the difference in the nomenclature. Note that the Bessho-Newman relation (\ref{eq:BN}) is a general form valid for any floating body, symmetric with respect to the $x$ axis, in a channel (or for an infinite array of such bodies). Considering $k<2\pi$, i.e. $\bar{q}=0$, and using the identities $\mathcal{A}_0^-=-\mathcal{A}_0^+$ (see \ref{eq:A0}), with $\mathcal{A}_0^+=|\mathcal{A}_0^+|e^{i\delta}$, and $T_0=1-R_0$ (see \ref{eq:T0}), (\ref{eq:BN}) becomes: $2 R_0-1=e^{2i\delta}$, which implies (\ref{eq:delta}). However, while with the general Bessho-Newman relation (\ref{eq:BN}) it is possible to obtain (\ref{eq:delta}) only in the domain $k<2\pi$, usage of the explicit forms (\ref{eq:A0}) and (\ref{eq:R0}), respectively for $\mathcal{A}_0^+$ and $R_0$, has allowed to extend (\ref{eq:delta}) to any wavenumber. Furthermore, by using (\ref{eq:T0}) and (\ref{eq:delta}), (\ref{eq:BN}) yields
\begin{equation}
\cos\delta=|\mathcal{A}_0^+|^{-1}\sum_{q=0}^{\bar{q}}\frac{\gamma_q}{\epsilon_q}\,\left|\mathcal{A}_q^+R_q\right|,
\label{eq:cosdelta}
\end{equation}
for any $k$. Expression (\ref{eq:cosdelta}) is a particular form of the Bessho-Newman relation, valid for a periodic array of flap-type converters under normally-incident waves. Note that for $k<2\pi$, i.e. $\bar{q}=0$, all the transverse modes are trapped near the array and (\ref{eq:cosdelta}) reduces to
\begin{equation}
\cos\delta=|R_0|,\quad k<2\pi.
\label{eq:cosdeltaR0}
\end{equation}
\subsection{Relation between $F$ and $\mathcal{A}_0^\pm$ (array Haskind relation)}
Consider the complex exciting torque (\ref{eq:F}) and the fundamental radiation coefficient (\ref{eq:A0}). Isolating the term $\alpha_{00}$ from (\ref{eq:A0}) and substituting it into (\ref{eq:F}), yields after some algebra
\begin{equation}
F=\pm2A_I\mathcal{A}_0^{\pm}C_g.
\label{eq:haskind}
\end{equation}
According to (\ref{eq:haskind}), the long-crested component of the radiated wave field is sufficient to obtain the exciting torque acting on each flap, for any value of $k$. Furthermore, since $A_I$ and $C_g$ are real numbers, (\ref{eq:haskind}) requires
\begin{equation}
F=|F|e^{i\delta}.
\label{eq:phaseF}
\end{equation}
Expression (\ref{eq:haskind}) can be transformed into physical variables via (\ref{eq:nondimvar}), thus giving
$$F'=2\rho g b'A_I'\mathcal{A}_0^+{'}C_g'.$$
The latter is similar in form to the well-known two-dimensional Haskind relation \cite{ME05} except for the factor $b'$, which represents the array spacing. Finally, note that (\ref{eq:haskind}) is an extension to intermediate water depth of Srokosz's equation (4.3) in \cite{SR80}.
\subsection{Relation between $F$ and $R_0$}
The relation between the exciting torque and the fundamental reflection coefficient can be easily obtained by isolating $\beta_0$ from (\ref{eq:R0}), substituting it in (\ref{eq:F}) together with (\ref{eq:Zn}) and developing the algebra, so that
\begin{equation}
F=2A_IR_0\frac{\tanh kh}{k}\left(h-c+\frac{\cosh kc-\cosh kh}{k \sinh kh}\right).
\label{eq:relFR0}
\end{equation}
Hence the exciting torque acting on each flap is related to the amplitude of the long-crested component of the reflected wave field, for any value of $k$.
\subsection{Relation between $R_0$ and $\mathcal{A}_0^+$}
By equating (\ref{eq:haskind}) and (\ref{eq:relFR0}) it is immediate to obtain
\begin{equation}
\frac{R_0}{\mathcal{A}_0^+}=\frac{kC_g}{\tanh kh \left(h-c+\frac{\cosh kc-\cosh kh}{k \sinh kh} \right)},
\label{eq:relR0A0p}
\end{equation}
valid for any $k$. Physically, (\ref{eq:relR0A0p}) measures the ratio between the reflective capacity of the system as an array of screens and the radiative capacity of the system as an array of wavemakers, oscillating at unison. In short waves, where the flaps are deemed to be operating \cite{HE10}, it is roughly $k\gg 1$, so that (\ref{eq:relR0A0p}) becomes
\begin{equation}
\frac{R_0}{\mathcal{A}_0^+}\simeq \frac{\sqrt{k}}{2\left(h-c-k^{-1}\right)},
\label{eq:approxex}
\end{equation}
as shown in figure \ref{fig:shortwav}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm, trim= 5cm 10cm 5cm 0cm]{shortwav.eps}
\end{center}
\caption{Ratio $R_0/\mathcal{A}_0^+$ (\ref{eq:relR0A0p}) versus non-dimensional wavenumber $k$ (solid line) and approximate expression (\ref{eq:approxex}) for large $k$ (dashed line). Parameters of the system are $b'=91.6\,\mathrm{m}$, $h'=10.9\,\mathrm{m}$, $c'=1.5\,\mathrm{m}$.}
\label{fig:shortwav}
\end{figure}
Since the ratio (\ref{eq:approxex}) is $O(k^{1/2})$, the diffractive phenomena occurring in the system dominate over the radiative ones in short waves. This suggests that the effects of diffraction are not to be neglected if an accurate description of the system dynamics is to be pursued.
\subsection{Relation between $\nu$ and $\mathcal{A}_0^+$}
Consider expression (\ref{eq:nu}), which defines the radiation damping $\nu$ for the reference plate. Isolating $\Im\left\lbrace\alpha_{00}\right\rbrace$ in (\ref{eq:A0}) and substituting it into (\ref{eq:nu}) yields, after some algebra,
\begin{equation}
\nu=2\Re\left\lbrace \mathcal{A}_0^+\right\rbrace \frac{\tanh kh}{k}\left(h-c+\frac{\cosh kc-\cosh kh}{k\sinh kh}\right),
\label{eq:relnua0}
\end{equation}
for any $k$. Incidentally, by isolating $\mathcal{A}_0^+$ in the array Haskind relation (\ref{eq:haskind}) and substituting it into (\ref{eq:relnua0}), the latter becomes
$$\nu=\Re\left\lbrace \frac{F}{A_I}\right\rbrace \frac{\tanh kh}{kC_g}\left(h-c+\frac{\cosh kc-\cosh kh}{k\sinh kh}\right),$$
which corresponds to expression (C3) of Renzi \& Dias \cite{RD12}. Note that (\ref{eq:relFR0}), (\ref{eq:relR0A0p}) and (\ref{eq:relnua0}) allow to obtain the exciting torque and the radiation damping - and consequently the optimum generated power (\ref{eq:Popt}) - directly from the fundamental reflection coefficient $R_0$. This is a peculiar property of the flap-type converter and does not hold in general for converters of different shape.
The above relations have been used to check the numerical calculations in this paper. In order to assess the accuracy of computations for a given equation of the form $\mathrm{l.h.s}=\mathrm{r.h.s.}$, the relative error
\begin{equation}
\epsilon=\frac{|\mathrm{l.h.s.}-\mathrm{r.h.s.}|}{|\mathrm{r.h.s}|}
\label{eq:error}
\end{equation}
is defined. For a typical system configuration (see \ref{sec:appD}), taking 40 array modes, 5 depth modes and 5 terms in the Chebyshev expansion, is sufficient to obtain a relative error of $O (10^{-16})$ in calculating the Haskind relation (\ref{eq:haskind}) and $O(10^{-15})$ in calculating the remaining relations (\ref{eq:relFR0})--(\ref{eq:relnua0}). Hence the method of solution based on the Green's theorem of \cite{RD12} reveals to be fast convergent and very efficient. In the following, the influence of the array aperture on the performance of the system is assessed, based on the relations found in this section.
\section{Performance evaluation}
\label{sec:paran}
Consider the optimum capture factor
\begin{equation}
C_F^{opt}=\frac{1}{4}\frac{|F|^2}{\nu A_I^2 C_g w},
\label{eq:CFopt}
\end{equation}
obtained by substituting the optimum power output (\ref{eq:Popt}) into (\ref{eq:CF}). By replacing $F$ with (\ref{eq:relFR0}), $\nu$ with (\ref{eq:relnua0}), and by performing some algebra, (\ref{eq:CFopt}) can be rewritten as
\begin{equation}
C_F^{opt}=\frac{1}{2}\frac{|R_0|}{(1-a)\cos\delta},
\label{eq:CFopt2}
\end{equation}
where $\delta$ is still the argument of $R_0$. According to (\ref{eq:CFopt2}), the performance of the array depends on the reflection coefficient magnitude and argument, which in turn are functions of the array aperture. Hence the solution of the scattering problem alone is sufficient to assess the performance of the system via (\ref{eq:CFopt2}). This result confirms that diffraction effects are fundamental in wave-power extraction from flap-type WECs. Therefore, the empiric criterion for which ``to absorb waves means to generate waves'' \cite{FA02}, valid for small floating bodies in the absence of diffraction, does not apply here in full. In the following, expression (\ref{eq:CFopt2}) will be validated against known theories in the small-gap and point-absorber limits. Then the maximum capture factor will be assessed.
\subsection{Small-gap limit}
In the limit $a\rightarrow0$, the flaps become joined to each other and the system is two-dimensional. In this case it is $R_0\rightarrow1$, because of complete reflection of the incident wave in the diffraction problem. Then it is straightforward to show that (\ref{eq:CFopt2}) becomes
\begin{equation}
C_F^{opt}\rightarrow\frac{1}{2},
\label{eq:CF2d}
\end{equation}
i.e. the capture factor coincides with the classical hydrodynamic efficiency for 2D devices \cite{ME05}.
\subsection{Point-absorber limit}
Consider now the limit $w=w'/b'\rightarrow 0$, for fixed array spacing $b'$. In this limit, the wavelength $\lambda=2\pi/k$ of the incident wave is much larger than the flap width, $\lambda\gg w$, and the interaction between the flaps is weak. Hence the results of the present theory can be compared to those of Budal \cite{BU77} and Srokosz \cite{SR80} for an infinite array of point absorbers. For such a system, the efficiency is assessed via the absorption length
\begin{equation}
L'=\frac{P}{\frac{1}{2} A_I^2 C_g}\,b'=C_F w',
\label{eq:abslen}
\end{equation}
which is the ratio between the power captured by the single device and that incident per unit wave crest length. For an array of converters the optimum absorption length is
\begin{equation}
L'_{opt}=l'_{opt}\,s,
\label{eq:abslen2}
\end{equation}
where $l'_{opt}$ is the optimum absorption length for an isolated body and $s$ is an interaction factor \cite{BU77, SR80}. When $w\ll \lambda $, each flap can be considered as a three-dimensional axisymmetric body, whose optimum absorption length for given wavelength of the incident wave is
\begin{equation}
l'_{opt}=\xi \frac{\lambda}{2\pi}\,b',
\label{eq:pointabs}
\end{equation}
where $\xi=1$ for heave and $\xi=2$ for surge (see \cite{EV76,HE10}). By substituting (\ref{eq:pointabs}) into (\ref{eq:abslen2}), then the latter into (\ref{eq:abslen}) and employing (\ref{eq:CFopt2}) for the optimum capture factor, the interaction factor becomes
\begin{equation}
s=\frac{k}{2\xi}\frac{|R_0|}{\cos\delta},
\label{eq:qfactor}
\end{equation}
where $\xi=2$, because in the point-absorber approximation the flap moves essentially in surge \cite{HE10}. Figure \ref{fig:qfactor} shows the plot of $s$ versus the non-dimensional wavenumber $k=k'b'$ for a typical configuration in which $w=0.05$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm, trim= 5cm 9cm 5cm 0cm]{qfactor.eps}
\end{center}
\caption{Interaction factor (\ref{eq:qfactor}) versus non-dimensional wavelength $k$ for the point-absorber approximation. Parameters of the system are $b'=91.6\,\mathrm{m}$, $w'=4.58\,\mathrm{m}$, $h'=10.9\,\mathrm{m}$, $c'=1.5\,\mathrm{m}$. The ratio $w=w'/b'=0.05$ justifies the use of the point-absorber approximation. The vertical dash-dotted lines show the resonant wavenumbers $k=2(\bar{q}+1)\pi$; the grey dashed line shows the linear behaviour $s=k/4$ for $k<2\pi$.}
\label{fig:qfactor}
\end{figure}
When $k< 2\pi$ (i.e. $\lambda> 1$) all the transverse modes are trapped and the behaviour of the curve is linear: $s\simeq k'b'/(2\xi)$. This agrees formally with the results shown by Budal \cite{BU77} and Srokosz \cite{SR80} for an infinite array of heaving point-absorbers, where $\xi=1$. When $k>2\pi$, i.e. $\lambda<1$, incomplete trapping of the transverse modes strongly modifies the behaviour of the curve. The interaction factor globally decreases, but spikes occur near the resonant wavenumbers $k=2(\bar{q}+1)\pi$. Note that this dynamics is different from that shown in \cite{SR80} for a system of heaving point absorbers. In the latter, $s$ drops to zero when $k\rightarrow2(\bar{q}+1)\pi$ from the left, so that the trapping of the transverse modes has a detrimental effect on the performance of the system \cite[see fig. 2]{SR80}. Here, instead, resonance produces local maxima of $s$ near $k=2(\bar{q}+1)\pi$ (see again figure \ref{fig:qfactor}) and therefore is beneficial in increasing the optimum efficiency, even away from complete trapping. This happens because the resonance of the transverse modes enhances the horizontal (surge) actions and drops the vertical (heave) loads \cite{CH94}. Therefore surging WECs benefit the most from the resonant mechanisms activating in an array configuration.
\subsection{Maximum capture factor}
Consider again the optimum capture factor, given by (\ref{eq:CFopt2}). Expression (\ref{eq:cosdelta}) shows that $|R_0|/\cos\delta$ has a unit upper limit (\ref{eq:cosdeltaR0}) when all the transverse modes are trapped near the array. This situation is the most favourable for energy extraction and yields the maximum capture factor attainable by an array of oscillating wave energy converters of given aperture $a$. Substitution of (\ref{eq:cosdeltaR0}) into (\ref{eq:CFopt2}) yields
\begin{equation}
C_F^{opt}= C_F^{max}=\frac{1}{2(1-a)},\quad k<2\pi
\label{eq:maxCF}
\end{equation}
for the maximum capture factor. Incidentally, note that (\ref{eq:maxCF}) correspond to Srokosz's \cite{SR80} maximum efficiency $E^{max}=wC_F^{max}$ for a symmetric floating body in a channel. Since $a<1$, the maximum capture factor (\ref{eq:maxCF}) for the array configuration is larger than the well-known limit value of $1/2$, obtained in the small-gap approximation $a\rightarrow 0$ (see expression \ref{eq:CF2d}). Hence the mutual interaction between the flaps, which is responsible for the trapping of energy near the array in the form of short-crested waves, can increase the capture factor of the system \cite{RD12}.
\section{Application to wave energy extraction}
\label{sec:waven}
In applications, the incident wave period $T'$ and wavelength $\lambda'$ are known, together with the flap width $w'$. The array aperture $a$ needs to be optimised so that the capture factor is maximum. Expression (\ref{eq:maxCF}) for the maximum capture factor would suggest to increase the aperture as much as possible, so that $a\rightarrow 1$ and consequently $C_F^{max}\rightarrow \infty$. However, (\ref{eq:maxCF}) is to be regarded as a theoretical upper limit. This is motivated by two reasons. First and most important, as $a\rightarrow 1$, then $w=1-a\rightarrow 0$. Now recall that $w=w'/b'$, being $w'$ the width of the single flap, which in practical applications is large. As a consequence, in order to have $w\rightarrow 0$, it must be $b'\rightarrow \infty$. In this limit, $\lambda=\lambda'/b'\rightarrow 0$, $k\rightarrow \infty$ and expression (\ref{eq:maxCF}) for the maximum capture factor is no longer valid. Physically, by increasing the array aperture $a$, the spatial period $b'$ increases so much, that the array is no longer able to trap all the transverse modes, resulting in more energy leakage. Second, recall that expression (\ref{eq:maxCF}) for the theoretical maximum capture factor is obtained under the assumption (\ref{eq:bodyres}), i.e. that the self-oscillation frequency of each flap is tuned to the frequency of the incoming waves. However, flap-type converters are usually designed to avoid this eventuality. At body resonance, the stroke of the flap would exceed by far the amplitude of the incident wave \cite{HE10}. It is then clear that this condition is undesirable and not compatible with the power take-off mechanism \cite{CR08}. Away from body resonance, the maximum values of $C_F$ attained are smaller than those predicted by (\ref{eq:maxCF}). This is due to the presence of the inertial terms at the denominator of $P$ (\ref{eq:pow}), which in turn reduce $C_F$ (\ref{eq:CF}).
The mathematical model of \S\ref{sec:model} is now applied to determine an optimisation criterion for the array aperture $a$, which maximises the power output of an array of flap-type WECs. The configuration investigated here is that of an infinite array of converters similar to Oyster 800\texttrademark\footnote{Oyster is a trademark of Aquamarine Power Limited.}. Each converter has a width $w'=26\,\mathrm{m}$ and is placed upon a foundation of height $c'=1.5\,\mathrm{m}$ from the bottom of the ocean; water depth is $h'=10.9\,\mathrm{m}$. Monochromatic incident waves of amplitude $A'_I=1\mathrm{m}$, period $T'=7\,\mathrm{s}$ (wavelength $\lambda'= 62\,\mathrm{m}$), representative on average of the wave climate off the west coast of Ireland \cite{R12}, are considered. For these parameters, several different layouts, from compact ($a=0.30$) to sparse ($a=0.95$), are analysed to determine the optimum array aperture $a_{opt}$. In each case the power $P'=\rho A{'}^2b{'}^{3/2}g^{3/2}P$ extracted by a single flap and the relevant capture factor $C_F$ (\ref{eq:CF}) are calculated with the mathematical model of \S\ref{sec:model}. Table \ref{tab:1} shows the selected values of $a$, the corresponding value of the array spacing $b'$, the generated power $P'$ in $\mathrm{kW}$, the capture factor $C_F$ and the theoretical maximum $C_F^{max}$ .
\begin{table}\centering
\begin{tabular}{c c c c c c c}
\hline
$a$ & $0.3$& $0.40$ & $0.50$ & $a_{opt}=0.58$ & $0.70$ & $0.95$ \\ \hline
$b'\,(\mathrm{m})$ & 37 & 43 & 52 & 62 &87 & 520\\\hline
$P'\,(\mathrm{kW})$ & 504 & 564 & 660 & 795 & 574 & 605\\ \hline
$C_F$ & 0.60 & 0.68 & 0.79 & 0.95 & 0.69 & 0.73\\ \hline
$C_F^{max}$ & 0.71 & 0.83 & 1 & 1.19 & (1.67) & (10)\\\hline
\end{tabular}
\caption{Array spatial period $b'$, power output $P'$ , capture factor $C_F$ and maximum theoretical capture factor $C_F^{max}$ for an infinite array of flap-type converters similar to Oyster 800\texttrademark. Different apertures are considered, from compact ($a=0.3$) to sparse ($a=0.95$). Calculations are made with the mathematical model of \S\ref{sec:model}.}
\label{tab:1}
\end{table}
The largest power output and capture factor are attained at the optimum configuration $a=a_{opt}=0.58$, which corresponds to $\lambda'=b'$ (i.e. $k=2\pi$), the trapping wavelength of the first transverse mode \cite{RD12}. For $a<a_{opt}$, $\lambda'>b'$ (i.e. $k<2\pi$) and all the transverse modes are perfectly trapped (see table \ref{tab:1}). However, since $a$ is small, the theoretical maximum (\ref{eq:maxCF}) sets a relatively small upper limit for $C_F$. By increasing $a$, $C_F^{max}$ increases and so does the actual capture factor $C_F$, until it reaches its maximum at $a=a_{opt}$. For $a>a_{opt}$ the theoretical limit $C_F^{max}$ still increases, while the actual capture factor $C_F$ decreases. This happens since in these cases $\lambda'<b'$ (i.e. $k>2\pi$) and complete trapping of the transverse modes is not possible, so that (\ref{eq:maxCF}) does not hold in practice. Energy leakage associated to the propagating transverse waves lowers the power absorption of the array well below the theoretical maximum values. In conclusion, the optimum aperture that maximises the capture factor is the one for which $\lambda'=b'$, i.e.
\begin{equation}
a_{opt}=1-\frac{w'}{\lambda'},
\end{equation}
which can be used as a preliminary design formula.
The theory exposed here reveals to be useful for the optimisation of the efficiency of an infinite array of flap-type wave energy converters in incident monochromatic waves of given period. Of course, in real seas superposition of different wave components must be considered. The power output may thus vary, depending on the coupling between the spectrum energy period and the torque peak period \cite{R12, CL12}. Further analysis is therefore necessary to obtain more accurate estimates of wave power generation in random seas. Finally, due to real sea bottom conditions, converters in array are likely to be deployed in a staggered configuration and in a finite number. Ongoing work is investigating the dynamics of a finite array of staggered converters and will be disclosed in the near future.
\section{Conclusions}
A periodic array of flap-type WECs has been analysed in this work by using the semi-analytical model of Renzi \& Dias \cite{RD12}. Asymptotic analysis in the far field has allowed to obtain new expressions for the radiation, reflection and transmission coefficients. Relations have been determined between the $0$th-mode coefficients and the hydrodynamic parameters of the system. Some of these relations constitute an extension to intermediate water of the previous results obtained by Srokosz \cite{SR80} for an array of floating bodies in deep water, while some others are peculiar to the flap-type converter. The efficiency of the system, evaluated via the capture factor, has been shown to depend on the reflection coefficient magnitude and argument, which in turn are functions of the array aperture. This result shows that diffraction effects are fundamental in wave-power extraction from flap-type WECs. Unlike a line of heaving buoys \cite{SR80}, an array of flap-type WECs can exploit the resonance of transverse modes to attain high capture factor levels, even when complete trapping of the transverse modes does not occur. The maximum capture factor is attained in the regime of complete trapping, for which the amount of energy available for extraction is the largest. Given the wave period and the flap width, the capture factor can be maximised by varying the spacing between the flaps, such that complete trapping of the transverse modes occurs. These results have been obtained under the assumptions that the fluid is inviscid and the flow is irrotational. Viscous effects and turbulent dissipations may reduce the values predicted here, especially near trapping frequencies (see \cite{CH94}). \\
This work was funded by Science Foundation Ireland (SFI) under the research project ``High-end computational modelling for wave energy systems''. Discussions with Prof. D.V. Evans and Dr X. B. Chen have been illuminating. Numerical data provision by Dr G. Bellotti and Mr A. Abdolali is gratefully acknowledged.
| train/arxiv |
BkiUdfPxaKPQoka4R7Xr | 5 | 1 | "\\section{Introduction}\n\nDuring the last ten years, much have been done about classification of s(...TRUNCATED) | train/arxiv |
BkiUfGk5qsFCiE6Ue_hi | 5 | 1 | "\\section*{Data Analysis Recipes:\\\\\n Products of multivariate Gaussians\\\\\n in Bayesian infe(...TRUNCATED) | train/arxiv |
BkiUbCk5qWTD6essZG9s | 5 | 1 | "\\section{Introduction}\\label{sec:Introduction}\n The pioneering works of Feynman \\cite{feynman19(...TRUNCATED) | train/arxiv |
BkiUfp7xaKgQZUmnaXGu | 5 | 1 | "\\section{INTRODUCTION}\n\nMagnetically induced ferroelectricity or electric control of magnetic or(...TRUNCATED) | train/arxiv |
BkiUdak5qX_Bpe9RFdgN | 5 | 1 | "\\section{Introduction}\n\\label{intro}\n\nMost of the known exoplanets orbit late-F, G, or early-K(...TRUNCATED) | train/arxiv |
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Top 30B token SlimPajama Subset selected by the Readability rater
This repository contains the dataset described in the paper Meta-rater: A Multi-dimensional Data Selection Method for Pre-training Language Models.
Code: https://github.com/opendatalab/Meta-rater
Dataset Description
This dataset contains the top 30B tokens from the SlimPajama-627B corpus, selected using the Readability dimension of the PRRC (Professionalism, Readability, Reasoning, Cleanliness) framework. Each document in this subset is scored and filtered by a ModernBERT-based rater fine-tuned to assess the clarity, coherence, and ease of understanding of the text.
- Source: SlimPajama-627B Annotated Dataset
- Selection: Top 30B tokens by PRRC-Readability score
- Quality metric: Readability (0–5 scale, see below)
- Annotation coverage: 100% of selected subset
Dataset Statistics
- Total tokens: 30B (subset of SlimPajama-627B)
- Selection method: Top-ranked by PRRC-Readability ModernBERT rater
- Domains: Same as SlimPajama (CommonCrawl, C4, GitHub, Books, ArXiv, Wikipedia, StackExchange)
- Annotation: Each document has a readability score (0–5)
Readability Quality Metric
Readability evaluates the clarity, coherence, and ease of understanding of the text. Higher scores indicate content that is clear, well-structured, and easy to follow, while lower scores reflect text that is difficult to comprehend due to poor structure, grammar, or vocabulary.
- 0–1: Significant issues with clarity or coherence; difficult to read
- 2–3: Generally clear but with some sections that are hard to understand
- 4–5: Very clear, coherent, and easy to read
Scores are assigned by a ModernBERT model fine-tuned on Llama-3.3-70B-Instruct annotations, as described in the Meta-rater paper.
Annotation Process
- Initial annotation: Llama-3.3-70B-Instruct rated 500k+ SlimPajama samples for readability
- Model training: ModernBERT fine-tuned on these annotations
- Scoring: All SlimPajama documents scored by ModernBERT; top 30B tokens selected
Citation
If you use this dataset, please cite:
@article{zhuang2025meta,
title={Meta-rater: A Multi-dimensional Data Selection Method for Pre-training Language Models},
author={Zhuang, Xinlin and Peng, Jiahui and Ma, Ren and Wang, Yinfan and Bai, Tianyi and Wei, Xingjian and Qiu, Jiantao and Zhang, Chi and Qian, Ying and He, Conghui},
journal={arXiv preprint arXiv:2504.14194},
year={2025}
}
License
This dataset is released under the same license as the original SlimPajama dataset. See the original SlimPajama repository for details.
Contact
- Project Lead: Ren Ma ([email protected])
- Corresponding Author: Conghui He ([email protected])
- Issues: GitHub Issues
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