arXiv subjects

David P. Woodruff

Publications and source records attributed to David P. Woodruff.

At least 163 records · Page 9Linked to original sources

arXiv2020

Revisiting the Sample Complexity of Sparse Spectrum Approximation of Gaussian Processes

We introduce a new scalable approximation for Gaussian processes with provable guarantees which hold simultaneously over its entire parameter space. Our approximation is obtained from an improved sample complexity analysis for sparse spectrum Gaussian processes (SSGPs). In particular, our analysis shows that under a certain data disentangling condition, an SSGP's prediction and model evidence (for training) can well-approximate those of a full GP with low sample complexity. We also develop a new auto-encoding algorithm that finds a latent space to disentangle latent input coordinates into well-separated clusters, which is amenable to our sample complexity analysis. We validate our proposed method on several benchmarks with promising results supporting our theoretical analysis.

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arXiv2020

Optimal $\ell_1$ Column Subset Selection and a Fast PTAS for Low Rank Approximation

We study the problem of entrywise $\ell_1$ low rank approximation. We give the first polynomial time column subset selection-based $\ell_1$ low rank approximation algorithm sampling $\tilde{O}(k)$ columns and achieving an $\tilde{O}(k^{1/2})$-approximation for any $k$, improving upon the previous best $\tilde{O}(k)$-approximation and matching a prior lower bound for column subset selection-based $\ell_1$-low rank approximation which holds for any $\text{poly}(k)$ number of columns. We extend our results to obtain tight upper and lower bounds for column subset selection-based $\ell_p$ low rank approximation for any $1 < p < 2$, closing a long line of work on this problem. We next give a $(1 + \varepsilon)$-approximation algorithm for entrywise $\ell_p$ low rank approximation, for $1 \leq p < 2$, that is not a column subset selection algorithm. First, we obtain an algorithm which, given a matrix $A \in \mathbb{R}^{n \times d}$, returns a rank-$k$ matrix $\hat{A}$ in $2^{\text{poly}(k/\varepsilon)} + \text{poly}(nd)$ running time such that: $$\|A - \hat{A}\|_p \leq (1 + \varepsilon) \cdot OPT + \frac{\varepsilon}{\text{poly}(k)}\|A\|_p$$ where $OPT = \min_{A_k \text{ rank }k} \|A - A_k\|_p$. Using this algorithm, in the same running time we give an algorithm which obtains error at most $(1 + \varepsilon) \cdot OPT$ and outputs a matrix of rank at most $3k$ -- these algorithms significantly improve upon all previous $(1 + \varepsilon)$- and $O(1)$-approximation algorithms for the $\ell_p$ low rank approximation problem, which required at least $n^{\text{poly}(k/\varepsilon)}$ or $n^{\text{poly}(k)}$ running time, and either required strong bit complexity assumptions (our algorithms do not) or had bicriteria rank $3k$. Finally, we show hardness results which nearly match our $2^{\text{poly}(k)} + \text{poly}(nd)$ running time and the above additive error guarantee.

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arXiv2020

WOR and $p$'s: Sketches for $\ell_p$-Sampling Without Replacement

Weighted sampling is a fundamental tool in data analysis and machine learning pipelines. Samples are used for efficient estimation of statistics or as sparse representations of the data. When weight distributions are skewed, as is often the case in practice, without-replacement (WOR) sampling is much more effective than with-replacement (WR) sampling: it provides a broader representation and higher accuracy for the same number of samples. We design novel composable sketches for WOR $\ell_p$ sampling, weighted sampling of keys according to a power $p\in[0,2]$ of their frequency (or for signed data, sum of updates). Our sketches have size that grows only linearly with the sample size. Our design is simple and practical, despite intricate analysis, and based on off-the-shelf use of widely implemented heavy hitters sketches such as CountSketch. Our method is the first to provide WOR sampling in the important regime of $p>1$ and the first to handle signed updates for $p>0$.

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arXiv2020

Near Input Sparsity Time Kernel Embeddings via Adaptive Sampling

To accelerate kernel methods, we propose a near input sparsity time algorithm for sampling the high-dimensional feature space implicitly defined by a kernel transformation. Our main contribution is an importance sampling method for subsampling the feature space of a degree $q$ tensoring of data points in almost input sparsity time, improving the recent oblivious sketching method of (Ahle et al., 2020) by a factor of $q^{5/2}/ε^2$. This leads to a subspace embedding for the polynomial kernel, as well as the Gaussian kernel, with a target dimension that is only linearly dependent on the statistical dimension of the kernel and in time which is only linearly dependent on the sparsity of the input dataset. We show how our subspace embedding bounds imply new statistical guarantees for kernel ridge regression. Furthermore, we empirically show that in large-scale regression tasks, our algorithm outperforms state-of-the-art kernel approximation methods.

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arXiv2020

Streaming Complexity of SVMs

We study the space complexity of solving the bias-regularized SVM problem in the streaming model. This is a classic supervised learning problem that has drawn lots of attention, including for developing fast algorithms for solving the problem approximately. One of the most widely used algorithms for approximately optimizing the SVM objective is Stochastic Gradient Descent (SGD), which requires only $O(\frac{1}{λε})$ random samples, and which immediately yields a streaming algorithm that uses $O(\frac{d}{λε})$ space. For related problems, better streaming algorithms are only known for smooth functions, unlike the SVM objective that we focus on in this work. We initiate an investigation of the space complexity for both finding an approximate optimum of this objective, and for the related ``point estimation'' problem of sketching the data set to evaluate the function value $F_λ$ on any query $(θ, b)$. We show that, for both problems, for dimensions $d=1,2$, one can obtain streaming algorithms with space polynomially smaller than $\frac{1}{λε}$, which is the complexity of SGD for strongly convex functions like the bias-regularized SVM, and which is known to be tight in general, even for $d=1$. We also prove polynomial lower bounds for both point estimation and optimization. In particular, for point estimation we obtain a tight bound of $Θ(1/\sqrtε)$ for $d=1$ and a nearly tight lower bound of $\widetildeΩ(d/ε^2)$ for $d = Ω( \log(1/ε))$. Finally, for optimization, we prove a $Ω(1/\sqrtε)$ lower bound for $d = Ω( \log(1/ε))$, and show similar bounds when $d$ is constant.

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arXiv2020

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ over a fixed field $\mathbb{F}$ for a specified pair of vectors $\boldsymbol{u},\boldsymbol{v} \in \mathbb{F}^n$. To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity.

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arXiv2020

Non-Adaptive Adaptive Sampling on Turnstile Streams

Adaptive sampling is a useful algorithmic tool for data summarization problems in the classical centralized setting, where the entire dataset is available to the single processor performing the computation. Adaptive sampling repeatedly selects rows of an underlying matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$, where $n\gg d$, with probabilities proportional to their distances to the subspace of the previously selected rows. Intuitively, adaptive sampling seems to be limited to trivial multi-pass algorithms in the streaming model of computation due to its inherently sequential nature of assigning sampling probabilities to each row only after the previous iteration is completed. Surprisingly, we show this is not the case by giving the first one-pass algorithms for adaptive sampling on turnstile streams and using space $\text{poly}(d,k,\log n)$, where $k$ is the number of adaptive sampling rounds to be performed. Our adaptive sampling procedure has a number of applications to various data summarization problems that either improve state-of-the-art or have only been previously studied in the more relaxed row-arrival model. We give the first relative-error algorithms for column subset selection, subspace approximation, projective clustering, and volume maximization on turnstile streams that use space sublinear in $n$. We complement our volume maximization algorithmic results with lower bounds that are tight up to lower order terms, even for multi-pass algorithms. By a similar construction, we also obtain lower bounds for volume maximization in the row-arrival model, which we match with competitive upper bounds. See paper for full abstract.

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arXiv2020

Low Rank Approximation with Entrywise $\ell_1$-Norm Error

We study the $\ell_1$-low rank approximation problem, where for a given $n \times d$ matrix $A$ and approximation factor $α\geq 1$, the goal is to output a rank-$k$ matrix $\widehat{A}$ for which $$\|A-\widehat{A}\|_1 \leq α\cdot \min_{\textrm{rank-}k\textrm{ matrices}~A'}\|A-A'\|_1,$$ where for an $n \times d$ matrix $C$, we let $\|C\|_1 = \sum_{i=1}^n \sum_{j=1}^d |C_{i,j}|$. This error measure is known to be more robust than the Frobenius norm in the presence of outliers and is indicated in models where Gaussian assumptions on the noise may not apply. The problem was shown to be NP-hard by Gillis and Vavasis and a number of heuristics have been proposed. It was asked in multiple places if there are any approximation algorithms. We give the first provable approximation algorithms for $\ell_1$-low rank approximation, showing that it is possible to achieve approximation factor $α= (\log d) \cdot \mathrm{poly}(k)$ in $\mathrm{nnz}(A) + (n+d) \mathrm{poly}(k)$ time, where $\mathrm{nnz}(A)$ denotes the number of non-zero entries of $A$. If $k$ is constant, we further improve the approximation ratio to $O(1)$ with a $\mathrm{poly}(nd)$-time algorithm. Under the Exponential Time Hypothesis, we show there is no $\mathrm{poly}(nd)$-time algorithm achieving a $(1+\frac{1}{\log^{1+γ}(nd)})$-approximation, for $γ> 0$ an arbitrarily small constant, even when $k = 1$. We give a number of additional results for $\ell_1$-low rank approximation: nearly tight upper and lower bounds for column subset selection, CUR decompositions, extensions to low rank approximation with respect to $\ell_p$-norms for $1 \leq p < 2$ and earthmover distance, low-communication distributed protocols and low-memory streaming algorithms, algorithms with limited randomness, and bicriteria algorithms. We also give a preliminary empirical evaluation.

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arXiv2020

Towards a Zero-One Law for Column Subset Selection

There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-$k$ matrix $B$ minimizing the sum of absolute values of differences to a given $n$-by-$n$ matrix $A$, $\min_{\textrm{rank-}k~B}\|A-B\|_1$, or more generally finding a rank-$k$ matrix $B$ which minimizes the sum of $p$-th powers of absolute values of differences, $\min_{\textrm{rank-}k~B}\|A-B\|_p^p$. Many of these algorithms are linear time columns subset selection algorithms, returning a subset of $\mathrm{poly}(k \log n)$ columns whose cost is no more than a $\mathrm{poly}(k)$ factor larger than the cost of the best rank-$k$ matrix. The above error measures are special cases of the following general entrywise low rank approximation problem: given an arbitrary function $g:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$, find a rank-$k$ matrix $B$ which minimizes $\|A-B\|_g = \sum_{i,j}g(A_{i,j}-B_{i,j})$. A natural question is which functions $g$ admit efficient approximation algorithms? Indeed, this is a central question of recent work studying generalized low rank models. In this work we give approximation algorithms for $\textit{every}$ function $g$ which is approximately monotone and satisfies an approximate triangle inequality, and we show both of these conditions are necessary. Further, our algorithm is efficient if the function $g$ admits an efficient approximate regression algorithm. Our approximation algorithms handle functions which are not even scale-invariant, such as the Huber loss function, which we show have very different structural properties than $\ell_p$-norms, e.g., one can show the lack of scale-invariance causes any column subset selection algorithm to provably require a $\sqrt{\log n}$ factor larger number of columns than $\ell_p$-norms; nevertheless we design the first efficient column subset selection algorithms for such error measures.

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arXiv2020

Average Case Column Subset Selection for Entrywise $\ell_1$-Norm Loss

We study the column subset selection problem with respect to the entrywise $\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{Ω(1)}$ number of columns to obtain a $(1+ε)$-approximation to the best entrywise $\ell_1$-norm low rank approximation of an $n \times n$ matrix. Nevertheless, we show that under certain minimal and realistic distributional settings, it is possible to obtain a $(1+ε)$-approximation with a nearly linear running time and poly$(k/ε)+O(k\log n)$ columns. Namely, we show that if the input matrix $A$ has the form $A = B + E$, where $B$ is an arbitrary rank-$k$ matrix, and $E$ is a matrix with i.i.d. entries drawn from any distribution $μ$ for which the $(1+γ)$-th moment exists, for an arbitrarily small constant $γ> 0$, then it is possible to obtain a $(1+ε)$-approximate column subset selection to the entrywise $\ell_1$-norm in nearly linear time. Conversely we show that if the first moment does not exist, then it is not possible to obtain a $(1+ε)$-approximate subset selection algorithm even if one chooses any $n^{o(1)}$ columns. This is the first algorithm of any kind for achieving a $(1+ε)$-approximation for entrywise $\ell_1$-norm loss low rank approximation.

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arXiv2020

Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $α$ be the cardinality of the largest independent set. Our goal is to estimate $α$ in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating $β$, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating $α$ and $β$. For unit-length intervals, we obtain a $(2+ε)$-approximation to $α$ and $β$ in poly$(\frac{\log(n)}ε)$ space. We also show a matching lower bound. Combined with the $3/2$-approximation for insertion-only streams by Cabello and Perez-Lanterno [CP15], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a $\left(\frac{8\sqrt{3}}π\right)$-approximation to $α$ and $β$ in poly$(\log(n), ε^{-1})$ space, which is closely related to the hexagonal circle packing constant. We provide algorithms for estimating $α$ for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating $α$ and $β$, where the parameter is the ratio of maximum to minimum interval length.

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arXiv2020

LSF-Join: Locality Sensitive Filtering for Distributed All-Pairs Set Similarity Under Skew

All-pairs set similarity is a widely used data mining task, even for large and high-dimensional datasets. Traditionally, similarity search has focused on discovering very similar pairs, for which a variety of efficient algorithms are known. However, recent work highlights the importance of finding pairs of sets with relatively small intersection sizes. For example, in a recommender system, two users may be alike even though their interests only overlap on a small percentage of items. In such systems, some dimensions are often highly skewed because they are very popular. Together these two properties render previous approaches infeasible for large input sizes. To address this problem, we present a new distributed algorithm, LSF-Join, for approximate all-pairs set similarity. The core of our algorithm is a randomized selection procedure based on Locality Sensitive Filtering. Our method deviates from prior approximate algorithms, which are based on Locality Sensitive Hashing. Theoretically, we show that LSF-Join efficiently finds most close pairs, even for small similarity thresholds and for skewed input sets. We prove guarantees on the communication, work, and maximum load of LSF-Join, and we also experimentally demonstrate its accuracy on multiple graphs.

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arXiv2020

Span Recovery for Deep Neural Networks with Applications to Input Obfuscation

The tremendous success of deep neural networks has motivated the need to better understand the fundamental properties of these networks, but many of the theoretical results proposed have only been for shallow networks. In this paper, we study an important primitive for understanding the meaningful input space of a deep network: span recovery. For $k<n$, let $\mathbf{A} \in \mathbb{R}^{k \times n}$ be the innermost weight matrix of an arbitrary feed forward neural network $M:\mathbb{R}^n \to \mathbb{R}$, so $M(x)$ can be written as $M(x) = σ(\mathbf{A} x)$, for some network $σ:\mathbb{R}^k \to \mathbb{R}$. The goal is then to recover the row span of $\mathbf{A}$ given only oracle access to the value of $M(x)$. We show that if $M$ is a multi-layered network with ReLU activation functions, then partial recovery is possible: namely, we can provably recover $k/2$ linearly independent vectors in the row span of $\mathbf{A}$ using poly$(n)$ non-adaptive queries to $M(x)$. Furthermore, if $M$ has differentiable activation functions, we demonstrate that full span recovery is possible even when the output is first passed through a sign or $0/1$ thresholding function; in this case our algorithm is adaptive. Empirically, we confirm that full span recovery is not always possible, but only for unrealistically thin layers. For reasonably wide networks, we obtain full span recovery on both random networks and networks trained on MNIST data. Furthermore, we demonstrate the utility of span recovery as an attack by inducing neural networks to misclassify data obfuscated by controlled random noise as sensical inputs.

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arXiv2019

Sublinear Time Numerical Linear Algebra for Structured Matrices

We show how to solve a number of problems in numerical linear algebra, such as least squares regression, $\ell_p$-regression for any $p \geq 1$, low rank approximation, and kernel regression, in time $T(A) \poly(\log(nd))$, where for a given input matrix $A \in \mathbb{R}^{n \times d}$, $T(A)$ is the time needed to compute $A\cdot y$ for an arbitrary vector $y \in \mathbb{R}^d$. Since $T(A) \leq O(\nnz(A))$, where $\nnz(A)$ denotes the number of non-zero entries of $A$, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, $T(A)$ can be much smaller than $\nnz(A)$, yielding significantly sublinear time algorithms. For example, in the overconstrained $(1+ε)$-approximate polynomial interpolation problem, $A$ is a Vandermonde matrix and $T(A) = O(n \log n)$; in this case our running time is $n \cdot \poly(\log n) + \poly(d/ε)$ and we recover the results of \cite{avron2013sketching} as a special case. For overconstrained autoregression, which is a common problem arising in dynamical systems, $T(A) = O(n \log n)$, and we immediately obtain $n \cdot \poly(\log n) + \poly(d/ε)$ time. For kernel autoregression, we significantly improve the running time of prior algorithms for general kernels. For the important case of autoregression with the polynomial kernel and arbitrary target vector $b\in\mathbb{R}^n$, we obtain even faster algorithms. Our algorithms show that, perhaps surprisingly, most of these optimization problems do not require much more time than that of a polylogarithmic number of matrix-vector multiplications.

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arXiv2019

Pseudo-deterministic Streaming

A pseudo-deterministic algorithm is a (randomized) algorithm which, when run multiple times on the same input, with high probability outputs the same result on all executions. Classic streaming algorithms, such as those for finding heavy hitters, approximate counting, $\ell_2$ approximation, finding a nonzero entry in a vector (for turnstile algorithms) are not pseudo-deterministic. For example, in the instance of finding a nonzero entry in a vector, for any known low-space algorithm $A$, there exists a stream $x$ so that running $A$ twice on $x$ (using different randomness) would with high probability result in two different entries as the output. In this work, we study whether it is inherent that these algorithms output different values on different executions. That is, we ask whether these problems have low-memory pseudo-deterministic algorithms. For instance, we show that there is no low-memory pseudo-deterministic algorithm for finding a nonzero entry in a vector (given in a turnstile fashion), and also that there is no low-dimensional pseudo-deterministic sketching algorithm for $\ell_2$ norm estimation. We also exhibit problems which do have low memory pseudo-deterministic algorithms but no low memory deterministic algorithm, such as outputting a nonzero row of a matrix, or outputting a basis for the row-span of a matrix. We also investigate multi-pseudo-deterministic algorithms: algorithms which with high probability output one of a few options. We show the first lower bounds for such algorithms. This implies that there are streaming problems such that every low space algorithm for the problem must have inputs where there are many valid outputs, all with a significant probability of being outputted.

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arXiv2019

Graph Spanners in the Message-Passing Model

Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been extensively studied. We study the problem of computing a graph spanner when the edges of the input graph are distributed across two or more sites in an arbitrary, possibly worst-case partition, and the goal is for the sites to minimize the communication used to output a spanner. We assume the message-passing model of communication, for which there is a point-to-point link between all pairs of sites as well as a coordinator who is responsible for producing the output. We stress that the subset of edges that each site has is not related to the network topology, which is fixed to be point-to-point. While this model has been extensively studied for related problems such as graph connectivity, it has not been systematically studied for graph spanners. We present the first tradeoffs for total communication versus the quality of the spanners computed, for two or more sites, as well as for additive and multiplicative notions of distortion. We show separations in the communication complexity when edges are allowed to occur on multiple sites, versus when each edge occurs on at most one site. We obtain nearly tight bounds (up to polylog factors) for the communication of additive $2$-spanners in both the with and without duplication models, multiplicative $(2k-1)$-spanners in the with duplication model, and multiplicative $3$ and $5$-spanners in the without duplication model. Our lower bound for multiplicative $3$-spanners employs biregular bipartite graphs rather than the usual Erdős girth conjecture graphs and may be of wider interest.

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arXiv2019

Perfect $L_p$ Sampling in a Data Stream

In this paper, we resolve the one-pass space complexity of $L_p$ sampling for $p \in (0,2)$. Given a stream of updates (insertions and deletions) to the coordinates of an underlying vector $f \in \mathbb{R}^n$, a perfect $L_p$ sampler must output an index $i$ with probability $|f_i|^p/\|f\|_p^p$, and is allowed to fail with some probability $δ$. So far, for $p > 0$ no algorithm has been shown to solve the problem exactly using $\text{poly}( \log n)$-bits of space. In 2010, Monemizadeh and Woodruff introduced an approximate $L_p$ sampler, which outputs $i$ with probability $(1 \pm ν)|f_i|^p /\|f\|_p^p$, using space polynomial in $ν^{-1}$ and $\log(n)$. The space complexity was later reduced by Jowhari, Sağlam, and Tardos to roughly $O(ν^{-p} \log^2 n \log δ^{-1})$ for $p \in (0,2)$, which tightly matches the $Ω(\log^2 n \log δ^{-1})$ lower bound in terms of $n$ and $δ$, but is loose in terms of $ν$. Given these nearly tight bounds, it is perhaps surprising that no lower bound exists in terms of $ν$---not even a bound of $Ω(ν^{-1})$ is known. In this paper, we explain this phenomenon by demonstrating the existence of an $O(\log^2 n \log δ^{-1})$-bit perfect $L_p$ sampler for $p \in (0,2)$. This shows that $ν$ need not factor into the space of an $L_p$ sampler, which closes the complexity of the problem for this range of $p$. For $p=2$, our bound is $O(\log^3 n \log δ^{-1})$-bits, which matches the prior best known upper bound in terms of $n,δ$, but has no dependence on $ν$. For $p<2$, our bound holds in the random oracle model, matching the lower bounds in that model. Moreover, we show that our algorithm can be derandomized with only a $O((\log \log n)^2)$ blow-up in the space (and no blow-up for $p=2$). Our derandomization technique is general, and can be used to derandomize a large class of linear sketches.

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arXiv2019

Querying a Matrix through Matrix-Vector Products

We consider algorithms with access to an unknown matrix $M\in\mathbb{F}^{n \times d}$ via matrix-vector products, namely, the algorithm chooses vectors $\mathbf{v}^1, \ldots, \mathbf{v}^q$, and observes $M\mathbf{v}^1,\ldots, M\mathbf{v}^q$. Here the $\mathbf{v}^i$ can be randomized as well as chosen adaptively as a function of $ M\mathbf{v}^1,\ldots,M\mathbf{v}^{i-1}$. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number $q$ of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix $M$; to compute the AND/OR/Parity of each column or row of $M$, to decide whether there are identical columns or rows in $M$ or whether $M$ is symmetric, diagonal, or unitary; or to compute whether a graph defined by $M$ is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

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