arXiv subjects

David P. Woodruff

Publications and source records attributed to David P. Woodruff.

At least 145 records · Page 8Linked to original sources

arXiv2021

Single Pass Entrywise-Transformed Low Rank Approximation

In applications such as natural language processing or computer vision, one is given a large $n \times d$ matrix $A = (a_{i,j})$ and would like to compute a matrix decomposition, e.g., a low rank approximation, of a function $f(A) = (f(a_{i,j}))$ applied entrywise to $A$. A very important special case is the likelihood function $f\left( A \right ) = \log{\left( \left| a_{ij}\right| +1\right)}$. A natural way to do this would be to simply apply $f$ to each entry of $A$, and then compute the matrix decomposition, but this requires storing all of $A$ as well as multiple passes over its entries. Recent work of Liang et al.\ shows how to find a rank-$k$ factorization to $f(A)$ for an $n \times n$ matrix $A$ using only $n \cdot \operatorname{poly}(ε^{-1}k\log n)$ words of memory, with overall error $10\|f(A)-[f(A)]_k\|_F^2 + \operatorname{poly}(ε/k) \|f(A)\|_{1,2}^2$, where $[f(A)]_k$ is the best rank-$k$ approximation to $f(A)$ and $\|f(A)\|_{1,2}^2$ is the square of the sum of Euclidean lengths of rows of $f(A)$. Their algorithm uses three passes over the entries of $A$. The authors pose the open question of obtaining an algorithm with $n \cdot \operatorname{poly}(ε^{-1}k\log n)$ words of memory using only a single pass over the entries of $A$. In this paper we resolve this open question, obtaining the first single-pass algorithm for this problem and for the same class of functions $f$ studied by Liang et al. Moreover, our error is $\|f(A)-[f(A)]_k\|_F^2 + \operatorname{poly}(ε/k) \|f(A)\|_F^2$, where $\|f(A)\|_F^2$ is the sum of squares of Euclidean lengths of rows of $f(A)$. Thus our error is significantly smaller, as it removes the factor of $10$ and also $\|f(A)\|_F^2 \leq \|f(A)\|_{1,2}^2$. We also give an algorithm for regression, pointing out an error in previous work, and empirically validate our results.

cs.DS↗

arXiv2021

In-Database Regression in Input Sparsity Time

Sketching is a powerful dimensionality reduction technique for accelerating algorithms for data analysis. A crucial step in sketching methods is to compute a subspace embedding (SE) for a large matrix $\mathbf{A} \in \mathbb{R}^{N \times d}$. SE's are the primary tool for obtaining extremely efficient solutions for many linear-algebraic tasks, such as least squares regression and low rank approximation. Computing an SE often requires an explicit representation of $\mathbf{A}$ and running time proportional to the size of $\mathbf{A}$. However, if $\mathbf{A}= \mathbf{T}_1 \Join \mathbf{T}_2 \Join \dots \Join \mathbf{T}_m$ is the result of a database join query on several smaller tables $\mathbf{T}_i \in \mathbb{R}^{n_i \times d_i}$, then this running time can be prohibitive, as $\mathbf{A}$ itself can have as many as $O(n_1 n_2 \cdots n_m)$ rows. In this work, we design subspace embeddings for database joins which can be computed significantly faster than computing the join. For the case of a two table join $\mathbf{A} = \mathbf{T}_1 \Join \mathbf{T}_2$ we give input-sparsity algorithms for computing subspace embeddings, with running time bounded by the number of non-zero entries in $\mathbf{T}_1,\mathbf{T}_2$. This results in input-sparsity time algorithms for high accuracy regression, significantly improving upon the running time of prior FAQ-based methods for regression. We extend our results to arbitrary joins for the ridge regression problem, also considerably improving the running time of prior methods. Empirically, we apply our method to real datasets and show that it is significantly faster than existing algorithms.

cs.DS↗

arXiv2021

Average-Case Communication Complexity of Statistical Problems

We study statistical problems, such as planted clique, its variants, and sparse principal component analysis in the context of average-case communication complexity. Our motivation is to understand the statistical-computational trade-offs in streaming, sketching, and query-based models. Communication complexity is the main tool for proving lower bounds in these models, yet many prior results do not hold in an average-case setting. We provide a general reduction method that preserves the input distribution for problems involving a random graph or matrix with planted structure. Then, we derive two-party and multi-party communication lower bounds for detecting or finding planted cliques, bipartite cliques, and related problems. As a consequence, we obtain new bounds on the query complexity in the edge-probe, vector-matrix-vector, matrix-vector, linear sketching, and $\mathbb{F}_2$-sketching models. Many of these results are nearly tight, and we use our techniques to provide simple proofs of some known lower bounds for the edge-probe model.

cs.CC↗

arXiv2021

Reduced-Rank Regression with Operator Norm Error

A common data analysis task is the reduced-rank regression problem: $$\min_{\textrm{rank-}k \ X} \|AX-B\|,$$ where $A \in \mathbb{R}^{n \times c}$ and $B \in \mathbb{R}^{n \times d}$ are given large matrices and $\|\cdot\|$ is some norm. Here the unknown matrix $X \in \mathbb{R}^{c \times d}$ is constrained to be of rank $k$ as it results in a significant parameter reduction of the solution when $c$ and $d$ are large. In the case of Frobenius norm error, there is a standard closed form solution to this problem and a fast algorithm to find a $(1+\varepsilon)$-approximate solution. However, for the important case of operator norm error, no closed form solution is known and the fastest known algorithms take singular value decomposition time. We give the first randomized algorithms for this problem running in time $$(nnz{(A)} + nnz{(B)} + c^2) \cdot k/\varepsilon^{1.5} + (n+d)k^2/ε+ c^ω,$$ up to a polylogarithmic factor involving condition numbers, matrix dimensions, and dependence on $1/\varepsilon$. Here $nnz{(M)}$ denotes the number of non-zero entries of a matrix $M$, and $ω$ is the exponent of matrix multiplication. As both (1) spectral low rank approximation ($A = B$) and (2) linear system solving ($n = c$ and $d = 1$) are special cases, our time cannot be improved by more than a $1/\varepsilon$ factor (up to polylogarithmic factors) without a major breakthrough in linear algebra. Interestingly, known techniques for low rank approximation, such as alternating minimization or sketch-and-solve, provably fail for this problem. Instead, our algorithm uses an existential characterization of a solution, together with Krylov methods, low degree polynomial approximation, and sketching-based preconditioning.

cs.DS↗

arXiv2021

Dimensionality Reduction for Sum-of-Distances Metric

We give a dimensionality reduction procedure to approximate the sum of distances of a given set of $n$ points in $R^d$ to any "shape" that lies in a $k$-dimensional subspace. Here, by "shape" we mean any set of points in $R^d$. Our algorithm takes an input in the form of an $n \times d$ matrix $A$, where each row of $A$ denotes a data point, and outputs a subspace $P$ of dimension $O(k^{3}/ε^6)$ such that the projections of each of the $n$ points onto the subspace $P$ and the distances of each of the points to the subspace $P$ are sufficient to obtain an $ε$-approximation to the sum of distances to any arbitrary shape that lies in a $k$-dimensional subspace of $R^d$. These include important problems such as $k$-median, $k$-subspace approximation, and $(j,l)$ subspace clustering with $j \cdot l \leq k$. Dimensionality reduction reduces the data storage requirement to $(n+d)k^{3}/ε^6$ from nnz$(A)$. Here nnz$(A)$ could potentially be as large as $nd$. Our algorithm runs in time nnz$(A)/ε^2 + (n+d)$poly$(k/ε)$, up to logarithmic factors. For dense matrices, where nnz$(A) \approx nd$, we give a faster algorithm, that runs in time $nd + (n+d)$poly$(k/ε)$ up to logarithmic factors. Our dimensionality reduction algorithm can also be used to obtain poly$(k/ε)$ size coresets for $k$-median and $(k,1)$-subspace approximation problems in polynomial time.

cs.DS↗

arXiv2021

Non-PSD Matrix Sketching with Applications to Regression and Optimization

A variety of dimensionality reduction techniques have been applied for computations involving large matrices. The underlying matrix is randomly compressed into a smaller one, while approximately retaining many of its original properties. As a result, much of the expensive computation can be performed on the small matrix. The sketching of positive semidefinite (PSD) matrices is well understood, but there are many applications where the related matrices are not PSD, including Hessian matrices in non-convex optimization and covariance matrices in regression applications involving complex numbers. In this paper, we present novel dimensionality reduction methods for non-PSD matrices, as well as their ``square-roots", which involve matrices with complex entries. We show how these techniques can be used for multiple downstream tasks. In particular, we show how to use the proposed matrix sketching techniques for both convex and non-convex optimization, $\ell_p$-regression for every $1 \leq p \leq \infty$, and vector-matrix-vector queries.

cs.LG↗

arXiv2021

Robust and Sample Optimal Algorithms for PSD Low-Rank Approximation

Recently, Musco and Woodruff (FOCS, 2017) showed that given an $n \times n$ positive semidefinite (PSD) matrix $A$, it is possible to compute a $(1+ε)$-approximate relative-error low-rank approximation to $A$ by querying $O(nk/ε^{2.5})$ entries of $A$ in time $O(nk/ε^{2.5} +n k^{ω-1}/ε^{2(ω-1)})$. They also showed that any relative-error low-rank approximation algorithm must query $Ω(nk/ε)$ entries of $A$, this gap has since remained open. Our main result is to resolve this question by obtaining an optimal algorithm that queries $O(nk/ε)$ entries of $A$ and outputs a relative-error low-rank approximation in $O(n(k/ε)^{ω-1})$ time. Note, our running time improves that of Musco and Woodruff, and matches the information-theoretic lower bound if the matrix-multiplication exponent $ω$ is $2$. We then extend our techniques to negative-type distance matrices. Bakshi and Woodruff (NeurIPS, 2018) showed a bi-criteria, relative-error low-rank approximation which queries $O(nk/ε^{2.5})$ entries and outputs a rank-$(k+4)$ matrix. We show that the bi-criteria guarantee is not necessary and obtain an $O(nk/ε)$ query algorithm, which is optimal. Our algorithm applies to all distance matrices that arise from metrics satisfying negative-type inequalities, including $\ell_1, \ell_2,$ spherical metrics and hypermetrics. Next, we introduce a new robust low-rank approximation model which captures PSD matrices that have been corrupted with noise. While a sample complexity lower bound precludes sublinear algorithms for arbitrary PSD matrices, we provide the first sublinear time and query algorithms when the corruption on the diagonal entries is bounded. As a special case, we show sample-optimal sublinear time algorithms for low-rank approximation of correlation matrices corrupted by noise.

cs.DS↗

arXiv2021

Hutch++: Optimal Stochastic Trace Estimation

We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \pm ε)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/ε)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/ε^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.

cs.DS↗

arXiv2021

Learning the Positions in CountSketch

We consider sketching algorithms which first quickly compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low rank approximation. In the learning-based sketching paradigm proposed by Indyk et al. [2019], the sketch matrix is found by choosing a random sparse matrix, e.g., the CountSketch, and then updating the values of the non-zero entries by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work we propose the first learning algorithm that also optimizes the locations of the non-zero entries. We show this algorithm gives better accuracy for low rank approximation than previous work, and apply it to other problems such as $k$-means clustering for the first time. We show that our algorithm is provably better in the spiked covariance model and for Zipfian matrices. We also show the importance of the sketch monotonicity property for combining learned sketches. Our empirical results show the importance of optimizing not only the values of the non-zero entries but also their positions.

cs.LG↗

arXiv2021

Approximation Algorithms for Sparse Principal Component Analysis

Principal component analysis (PCA) is a widely used dimension reduction technique in machine learning and multivariate statistics. To improve the interpretability of PCA, various approaches to obtain sparse principal direction loadings have been proposed, which are termed Sparse Principal Component Analysis (SPCA). In this paper, we present thresholding as a provably accurate, polynomial time, approximation algorithm for the SPCA problem, without imposing any restrictive assumptions on the input covariance matrix. Our first thresholding algorithm using the Singular Value Decomposition is conceptually simple; is faster than current state-of-the-art; and performs well in practice. On the negative side, our (novel) theoretical bounds do not accurately predict the strong practical performance of this approach. The second algorithm solves a well-known semidefinite programming relaxation and then uses a novel, two step, deterministic thresholding scheme to compute a sparse principal vector. It works very well in practice and, remarkably, this solid practical performance is accurately predicted by our theoretical bounds, which bridge the theory-practice gap better than current state-of-the-art.

cs.DS↗

arXiv2021

A Simple Proof of a New Set Disjointness with Applications to Data Streams

The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are $k$ players with subsets $S^1, \ldots, S^k$, each drawn from $\{1, 2, \ldots, n\}$, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element $j$ occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is $Ω(n/k)$. We observe for most applications, it instead suffices to look at what we call the ``mostly'' set disjointness problem, which changes case (2) to say there is a unique element $j$ occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an $Ω(n/k)$ randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Using this we show several new results for data streams: \begin{itemize} \item for $\ell_2$-Heavy Hitters, any $O(1)$-pass streaming algorithm in the insertion-only model for detecting if an $\eps$-$\ell_2$-heavy hitter exists requires $\min(\frac{1}{\eps^2}\log \frac{\eps^2n}δ, \frac{1}{\eps}n^{1/2})$ bits of memory, which is optimal up to a $\log n$ factor. For deterministic algorithms and constant $\eps$, this gives an $Ω(n^{1/2})$ lower bound, improving the prior $Ω(\log n)$ lower bound. We also obtain lower bounds for Zipfian distributions. \item for $\ell_p$-Estimation, $p > 2$, we show an $O(1)$-pass $Ω(n^{1-2/p} \log(1/δ))$ bit lower bound for outputting an $O(1)$-approximation with probability $1-δ$, in the insertion-only model. This is optimal, and the best previous lower bound was $Ω(n^{1-2/p} + \log(1/δ))$. \end{itemize}

cs.DS↗

arXiv2021

Learning a Latent Simplex in Input-Sparsity Time

We consider the problem of learning a latent $k$-vertex simplex $K\subset\mathbb{R}^d$, given access to $A\in\mathbb{R}^{d\times n}$, which can be viewed as a data matrix with $n$ points that are obtained by randomly perturbing latent points in the simplex $K$ (potentially beyond $K$). A large class of latent variable models, such as adversarial clustering, mixed membership stochastic block models, and topic models can be cast as learning a latent simplex. Bhattacharyya and Kannan (SODA, 2020) give an algorithm for learning such a latent simplex in time roughly $O(k\cdot\textrm{nnz}(A))$, where $\textrm{nnz}(A)$ is the number of non-zeros in $A$. We show that the dependence on $k$ in the running time is unnecessary given a natural assumption about the mass of the top $k$ singular values of $A$, which holds in many of these applications. Further, we show this assumption is necessary, as otherwise an algorithm for learning a latent simplex would imply an algorithmic breakthrough for spectral low rank approximation. At a high level, Bhattacharyya and Kannan provide an adaptive algorithm that makes $k$ matrix-vector product queries to $A$ and each query is a function of all queries preceding it. Since each matrix-vector product requires $\textrm{nnz}(A)$ time, their overall running time appears unavoidable. Instead, we obtain a low-rank approximation to $A$ in input-sparsity time and show that the column space thus obtained has small $\sinΘ$ (angular) distance to the right top-$k$ singular space of $A$. Our algorithm then selects $k$ points in the low-rank subspace with the largest inner product with $k$ carefully chosen random vectors. By working in the low-rank subspace, we avoid reading the entire matrix in each iteration and thus circumvent the $Θ(k\cdot\textrm{nnz}(A))$ running time.

cs.LG↗

arXiv2021

How to reduce dimension with PCA and random projections?

In our "big data" age, the size and complexity of data is steadily increasing. Methods for dimension reduction are ever more popular and useful. Two distinct types of dimension reduction are "data-oblivious" methods such as random projections and sketching, and "data-aware" methods such as principal component analysis (PCA). Both have their strengths, such as speed for random projections, and data-adaptivity for PCA. In this work, we study how to combine them to get the best of both. We study "sketch and solve" methods that take a random projection (or sketch) first, and compute PCA after. We compute the performance of several popular sketching methods (random iid projections, random sampling, subsampled Hadamard transform, count sketch, etc) in a general "signal-plus-noise" (or spiked) data model. Compared to well-known works, our results (1) give asymptotically exact results, and (2) apply when the signal components are only slightly above the noise, but the projection dimension is non-negligible. We also study stronger signals allowing more general covariance structures. We find that (a) signal strength decreases under projection in a delicate way depending on the structure of the data and the sketching method, (b) orthogonal projections are more accurate, (c) randomization does not hurt too much, due to concentration of measure, (d) count sketch can be improved by a normalization method. Our results have implications for statistical learning and data analysis. We also illustrate that the results are highly accurate in simulations and in analyzing empirical data.

math.ST↗

arXiv2021

On Coresets for Logistic Regression

Coresets are one of the central methods to facilitate the analysis of large data sets. We continue a recent line of research applying the theory of coresets to logistic regression. First, we show a negative result, namely, that no strongly sublinear sized coresets exist for logistic regression. To deal with intractable worst-case instances we introduce a complexity measure $μ(X)$, which quantifies the hardness of compressing a data set for logistic regression. $μ(X)$ has an intuitive statistical interpretation that may be of independent interest. For data sets with bounded $μ(X)$-complexity, we show that a novel sensitivity sampling scheme produces the first provably sublinear $(1\pm\varepsilon)$-coreset. We illustrate the performance of our method by comparing to uniform sampling as well as to state of the art methods in the area. The experiments are conducted on real world benchmark data for logistic regression.

cs.DS↗

arXiv2021

A PTAS for $\ell_p$-Low Rank Approximation

A number of recent works have studied algorithms for entrywise $\ell_p$-low rank approximation, namely, algorithms which given an $n \times d$ matrix $A$ (with $n \geq d$), output a rank-$k$ matrix $B$ minimizing $\|A-B\|_p^p=\sum_{i,j}|A_{i,j}-B_{i,j}|^p$ when $p > 0$; and $\|A-B\|_0=\sum_{i,j}[A_{i,j}\neq B_{i,j}]$ for $p=0$. On the algorithmic side, for $p \in (0,2)$, we give the first $(1+ε)$-approximation algorithm running in time $n^{\text{poly}(k/ε)}$. Further, for $p = 0$, we give the first almost-linear time approximation scheme for what we call the Generalized Binary $\ell_0$-Rank-$k$ problem. Our algorithm computes $(1+ε)$-approximation in time $(1/ε)^{2^{O(k)}/ε^{2}} \cdot nd^{1+o(1)}$. On the hardness of approximation side, for $p \in (1,2)$, assuming the Small Set Expansion Hypothesis and the Exponential Time Hypothesis (ETH), we show that there exists $δ:= δ(α) > 0$ such that the entrywise $\ell_p$-Rank-$k$ problem has no $α$-approximation algorithm running in time $2^{k^δ}$.

cs.DS↗

arXiv2021

Subspace exploration: Bounds on Projected Frequency Estimation

Given an $n \times d$ dimensional dataset $A$, a projection query specifies a subset $C \subseteq [d]$ of columns which yields a new $n \times |C|$ array. We study the space complexity of computing data analysis functions over such subspaces, including heavy hitters and norms, when the subspaces are revealed only after observing the data. We show that this important class of problems is typically hard: for many problems, we show $2^{Ω(d)}$ lower bounds. However, we present upper bounds which demonstrate space dependency better than $2^d$. That is, for $c,c' \in (0,1)$ and a parameter $N=2^d$ an $N^c$-approximation can be obtained in space $\min(N^{c'},n)$, showing that it is possible to improve on the naïve approach of keeping information for all $2^d$ subsets of $d$ columns. Our results are based on careful constructions of instances using coding theory and novel combinatorial reductions that exhibit such space-approximation tradeoffs.

cs.DS↗

arXiv2020

How to Fake Multiply by a Gaussian Matrix

Have you ever wanted to multiply an $n \times d$ matrix $X$, with $n \gg d$, on the left by an $m \times n$ matrix $\tilde G$ of i.i.d. Gaussian random variables, but could not afford to do it because it was too slow? In this work we propose a new randomized $m \times n$ matrix $T$, for which one can compute $T \cdot X$ in only $O(\text{nnz}(X)) + \tilde O(m^2 \cdot d^{3})$ time, for which the total variation distance between the distributions $T \cdot X$ and $\tilde G \cdot X$ is as small as desired, i.e., less than any positive constant. Here $\text{nnz}(X)$ denotes the number of non-zero entries of $X$. Assuming $\text{nnz}(X) \gg m^2 \cdot d^{3}$, this is a significant savings over the naïve $O(\text{nnz}(X) m)$ time to compute $\tilde G \cdot X$. Moreover, since the total variation distance is small, we can provably use $T \cdot X$ in place of $\tilde G \cdot X$ in any application and have the same guarantees as if we were using $\tilde G \cdot X$, up to a small positive constant in error probability. We apply this transform to nonnegative matrix factorization (NMF) and support vector machines (SVM).

cs.DS↗

arXiv2020

Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation

In $masked\ low-rank\ approximation$, one is given $A \in \mathbb{R}^{n \times n}$ and binary mask matrix $W \in \{0,1\}^{n \times n}$. The goal is to find a rank-$k$ matrix $L$ for which: $$cost(L) = \sum_{i=1}^{n} \sum_{j = 1}^{n} W_{i,j} \cdot (A_{i,j} - L_{i,j} )^2 \leq OPT + ε\|A\|_F^2 ,$$ where $OPT = \min_{rank-k\ \hat{L}} cost(\hat L)$ and $ε$ is a given error parameter. Depending on the choice of $W$, this problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, and many problems. Many of these problems are NP-hard, and while some algorithms with provable guarantees are known, they either 1) run in time $n^{Ω(k^2/ε)}$ or 2) make strong assumptions, e.g., that $A$ is incoherent or that $W$ is random. In this work, we show that a common polynomial time heuristic, which simply sets $A$ to $0$ where $W$ is $0$, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank $k' > k$ depending on the $public\ coin\ partition\ number$ of $W$, the heuristic outputs rank-$k'$ $L$ with cost$(L) \leq OPT + ε\|A\|_F^2$. This partition number is in turn bounded by the $randomized\ communication\ complexity$ of $W$, when interpreted as a two-player communication matrix. For many important examples of masked low-rank approximation, including all those listed above, this result yields bicriteria approximation guarantees with $k' = k \cdot poly(\log n/ε)$. Further, we show that different models of communication yield algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.

cs.DS↗