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David P. Woodruff

Publications and source records attributed to David P. Woodruff.

At least 199 records · Page 11Linked to original sources
arXiv2018

Leveraging Well-Conditioned Bases: Streaming \& Distributed Summaries in Minkowski $p$-Norms

Work on approximate linear algebra has led to efficient distributed and streaming algorithms for problems such as approximate matrix multiplication, low rank approximation, and regression, primarily for the Euclidean norm $\ell_2$. We study other $\ell_p$ norms, which are more robust for $p < 2$, and can be used to find outliers for $p > 2$. Unlike previous algorithms for such norms, we give algorithms that are (1) deterministic, (2) work simultaneously for every $p \geq 1$, including $p = \infty$, and (3) can be implemented in both distributed and streaming environments. We apply our results to $\ell_p$-regression, entrywise $\ell_1$-low rank approximation, and approximate matrix multiplication.

cs.DS↗
arXiv2018

Distributed Statistical Estimation of Matrix Products with Applications

We consider statistical estimations of a matrix product over the integers in a distributed setting, where we have two parties Alice and Bob; Alice holds a matrix $A$ and Bob holds a matrix $B$, and they want to estimate statistics of $A \cdot B$. We focus on the well-studied $\ell_p$-norm, distinct elements ($p = 0$), $\ell_0$-sampling, and heavy hitter problems. The goal is to minimize both the communication cost and the number of rounds of communication. This problem is closely related to the fundamental set-intersection join problem in databases: when $p = 0$ the problem corresponds to the size of the set-intersection join. When $p = \infty$ the output is simply the pair of sets with the maximum intersection size. When $p = 1$ the problem corresponds to the size of the corresponding natural join. We also consider the heavy hitters problem which corresponds to finding the pairs of sets with intersection size above a certain threshold, and the problem of sampling an intersecting pair of sets uniformly at random.

cs.DS↗
arXiv2018

On Sketching the $q$ to $p$ norms

We initiate the study of data dimensionality reduction, or sketching, for the $q\to p$ norms. Given an $n \times d$ matrix $A$, the $q\to p$ norm, denoted $\|A\|_{q \to p} = \sup_{x \in \mathbb{R}^d \backslash \vec{0}} \frac{\|Ax\|_p}{\|x\|_q}$, is a natural generalization of several matrix and vector norms studied in the data stream and sketching models, with applications to datamining, hardness of approximation, and oblivious routing. We say a distribution $S$ on random matrices $L \in \mathbb{R}^{nd} \rightarrow \mathbb{R}^k$ is a $(k,α)$-sketching family if from $L(A)$, one can approximate $\|A\|_{q \to p}$ up to a factor $α$ with constant probability. We provide upper and lower bounds on the sketching dimension $k$ for every $p, q \in [1, \infty]$, and in a number of cases our bounds are tight. While we mostly focus on constant $α$, we also consider large approximation factors $α$, as well as other variants of the problem such as when $A$ has low rank.

cs.DS↗
arXiv2018

High Probability Frequency Moment Sketches

We consider the problem of sketching the $p$-th frequency moment of a vector, $p>2$, with multiplicative error at most $1\pm ε$ and \emph{with high confidence} $1-δ$. Despite the long sequence of work on this problem, tight bounds on this quantity are only known for constant $δ$. While one can obtain an upper bound with error probability $δ$ by repeating a sketching algorithm with constant error probability $O(\log(1/δ))$ times in parallel, and taking the median of the outputs, we show this is a suboptimal algorithm! Namely, we show optimal upper and lower bounds of $Θ(n^{1-2/p} \log(1/δ) + n^{1-2/p} \log^{2/p} (1/δ) \log n)$ on the sketching dimension, for any constant approximation. Our result should be contrasted with results for estimating frequency moments for $1 \leq p \leq 2$, for which we show the optimal algorithm for general $δ$ is obtained by repeating the optimal algorithm for constant error probability $O(\log(1/δ))$ times and taking the median output. We also obtain a matching lower bound for this problem, up to constant factors.

cs.DS↗
arXiv2018

Matrix Completion and Related Problems via Strong Duality

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.

cs.DS↗
arXiv2018

Improved Algorithms for Adaptive Compressed Sensing

In the problem of adaptive compressed sensing, one wants to estimate an approximately $k$-sparse vector $x\in\mathbb{R}^n$ from $m$ linear measurements $A_1 x, A_2 x,\ldots, A_m x$, where $A_i$ can be chosen based on the outcomes $A_1 x,\ldots, A_{i-1} x$ of previous measurements. The goal is to output a vector $\hat{x}$ for which $$\|x-\hat{x}\|_p \le C \cdot \min_{k\text{-sparse } x'} \|x-x'\|_q\,$$ with probability at least $2/3$, where $C > 0$ is an approximation factor. Indyk, Price and Woodruff (FOCS'11) gave an algorithm for $p=q=2$ for $C = 1+ε$ with $\Oh((k/ε) \loglog (n/k))$ measurements and $\Oh(\log^*(k) \loglog (n))$ rounds of adaptivity. We first improve their bounds, obtaining a scheme with $\Oh(k \cdot \loglog (n/k) +(k/ε) \cdot \loglog(1/ε))$ measurements and $\Oh(\log^*(k) \loglog (n))$ rounds, as well as a scheme with $\Oh((k/ε) \cdot \loglog (n\log (n/k)))$ measurements and an optimal $\Oh(\loglog (n))$ rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for $(p,p)$ for every $0 < p < 2$. We show that the improvement from $O(k \log(n/k))$ measurements to $O(k \log \log (n/k))$ measurements in the adaptive setting can persist with a better $ε$-dependence for other values of $p$ and $q$. For example, when $(p,q) = (1,1)$, we obtain $O(\frac{k}{\sqrtε} \cdot \log \log n \log^3 (\frac{1}ε))$ measurements.

cs.DS↗
arXiv2018

Tight Bounds for $\ell_p$ Oblivious Subspace Embeddings

An $\ell_p$ oblivious subspace embedding is a distribution over $r \times n$ matrices $Π$ such that for any fixed $n \times d$ matrix $A$, $$\Pr_Π[\textrm{for all }x, \ \|Ax\|_p \leq \|ΠAx\|_p \leq κ\|Ax\|_p] \geq 9/10,$$ where $r$ is the dimension of the embedding, $κ$ is the distortion of the embedding, and for an $n$-dimensional vector $y$, $\|y\|_p$ is the $\ell_p$-norm. Another important property is the sparsity of $Π$, that is, the maximum number of non-zero entries per column, as this determines the running time of computing $Π\cdot A$. While for $p = 2$ there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparisty, for the important case of $1 \leq p < 2$, much less was known. In this paper we obtain nearly optimal tradeoffs for $\ell_p$ oblivious subspace embeddings for every $1 \leq p < 2$. We show for every $1 \leq p < 2$, any oblivious subspace embedding with dimension $r$ has distortion $κ= Ω\left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \cdot \log^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).$ When $r = \mathrm{poly}(d)$ in applications, this gives a $κ= Ω(d^{1/p}\log^{-2/p} d)$ lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for $p = 1$ and the oblivious subspace embedding of Meng and Mahoney (STOC, 2013) for $1 < p < 2$ are optimal up to $\mathrm{poly}(\log(d))$ factors. We also give sparse oblivious subspace embeddings for every $1 \leq p < 2$ which are optimal in dimension and distortion, up to $\mathrm{poly}(\log d)$ factors. Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise $\ell_p$ low rank approximation. Our results give improved algorithms for these applications.

cs.DS↗
arXiv2018

Relative Error Tensor Low Rank Approximation

We consider relative error low rank approximation of $tensors$ with respect to the Frobenius norm: given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT, where OPT $= \inf_{\textrm{rank-}k~A'} \|A-A'\|_F^2$. Despite the success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-$k$ tensor $A_k$ achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these issues via (1) bicriteria and (2) parameterized complexity solutions: (1) We give an algorithm which outputs a rank $k' = O((k/ε)^{q-1})$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT in $nnz(A) + n \cdot \textrm{poly}(k/ε)$ time in the real RAM model. Here $nnz(A)$ is the number of non-zero entries in $A$. (2) We give an algorithm for any $δ>0$ which outputs a rank $k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT and runs in $ ( nnz(A) + n \cdot \textrm{poly}(k/ε) + \exp(k^2/ε) ) \cdot n^δ$ time in the unit cost RAM model. For outputting a rank-$k$ tensor, or even a bicriteria solution with rank-$Ck$ for a certain constant $C > 1$, we show a $2^{Ω(k^{1-o(1)})}$ time lower bound under the Exponential Time Hypothesis. Our results give the first relative error low rank approximations for tensors for a large number of robust error measures for which nothing was known, as well as column row and tube subset selection. We also obtain new results for matrices, such as $nnz(A)$-time CUR decompositions, improving previous $nnz(A)\log n$-time algorithms, which may be of independent interest.

cs.DS↗
arXiv2018

Data Streams with Bounded Deletions

Two prevalent models in the data stream literature are the insertion-only and turnstile models. Unfortunately, many important streaming problems require a $Θ(\log(n))$ multiplicative factor more space for turnstile streams than for insertion-only streams. This complexity gap often arises because the underlying frequency vector $f$ is very close to $0$, after accounting for all insertions and deletions to items. Signal detection in such streams is difficult, given the large number of deletions. In this work, we propose an intermediate model which, given a parameter $α\geq 1$, lower bounds the norm $\|f\|_p$ by a $1/α$-fraction of the $L_p$ mass of the stream had all updates been positive. Here, for a vector $f$, $\|f\|_p = \left (\sum_{i=1}^n |f_i|^p \right )^{1/p}$, and the value of $p$ we choose depends on the application. This gives a fluid medium between insertion only streams (with $α= 1$), and turnstile streams (with $α= \text{poly}(n)$), and allows for analysis in terms of $α$. We show that for streams with this $α$-property, for many fundamental streaming problems we can replace a $O(\log(n))$ factor in the space usage for algorithms in the turnstile model with a $O(\log(α))$ factor. This is true for identifying heavy hitters, inner product estimation, $L_0$ estimation, $L_1$ estimation, $L_1$ sampling, and support sampling. For each problem, we give matching or nearly matching lower bounds for $α$-property streams. We note that in practice, many important turnstile data streams are in fact $α$-property streams for small values of $α$. For such applications, our results represent significant improvements in efficiency for all the aforementioned problems.

cs.DS↗
arXiv2017

Sketching for Kronecker Product Regression and P-splines

TensorSketch is an oblivious linear sketch introduced in Pagh'13 and later used in Pham, Pagh'13 in the context of SVMs for polynomial kernels. It was shown in Avron, Nguyen, Woodruff'14 that TensorSketch provides a subspace embedding, and therefore can be used for canonical correlation analysis, low rank approximation, and principal component regression for the polynomial kernel. We take TensorSketch outside of the context of polynomials kernels, and show its utility in applications in which the underlying design matrix is a Kronecker product of smaller matrices. This allows us to solve Kronecker product regression and non-negative Kronecker product regression, as well as regularized spline regression. Our main technical result is then in extending TensorSketch to other norms. That is, TensorSketch only provides input sparsity time for Kronecker product regression with respect to the $2$-norm. We show how to solve Kronecker product regression with respect to the $1$-norm in time sublinear in the time required for computing the Kronecker product, as well as for more general $p$-norms.

cs.DS↗
arXiv2017

BPTree: an $\ell_2$ heavy hitters algorithm using constant memory

The task of finding heavy hitters is one of the best known and well studied problems in the area of data streams. One is given a list $i_1,i_2,\ldots,i_m\in[n]$ and the goal is to identify the items among $[n]$ that appear frequently in the list. In sub-polynomial space, the strongest guarantee available is the $\ell_2$ guarantee, which requires finding all items that occur at least $ε\|f\|_2$ times in the stream, where the vector $f\in\mathbb{R}^n$ is the count histogram of the stream with $i$th coordinate equal to the number of times~$i$ appears $f_i:=\#\{j\in[m]:i_j=i\}$. The first algorithm to achieve the $\ell_2$ guarantee was the CountSketch of [CCF04], which requires $O(ε^{-2}\log n)$ words of memory and $O(\log n)$ update time and is known to be space-optimal if the stream allows for deletions. The recent work of [BCIW16] gave an improved algorithm for insertion-only streams, using only $O(ε^{-2}\logε^{-1}\log\log n)$ words of memory. In this work, we give an algorithm \bptree for $\ell_2$ heavy hitters in insertion-only streams that achieves $O(ε^{-2}\logε^{-1})$ words of memory and $O(\logε^{-1})$ update time, which is the optimal dependence on $n$ and $m$. In addition, we describe an algorithm for tracking $\|f\|_2$ at all times with $O(ε^{-2})$ memory and update time. Our analyses rely on bounding the expected supremum of a Bernoulli process involving Rademachers with limited independence, which we accomplish via a Dudley-like chaining argument that may have applications elsewhere.

cs.DS↗
arXiv2017

Is Input Sparsity Time Possible for Kernel Low-Rank Approximation?

Low-rank approximation is a common tool used to accelerate kernel methods: the $n \times n$ kernel matrix $K$ is approximated via a rank-$k$ matrix $\tilde K$ which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error $k$-rank approximation to $K$ is at least as difficult as multiplying the input data matrix $A \in \mathbb{R}^{n \times d}$ by an arbitrary matrix $C \in \mathbb{R}^{d \times k}$. Barring a breakthrough in fast matrix multiplication, when $k$ is not too large, this requires $Ω(nnz(A)k)$ time where $nnz(A)$ is the number of non-zeros in $A$. This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16,MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to $O(nnz(A))$ input sparsity runtimes. At the same time there is hope: we show for the first time that $O(nnz(A))$ time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset.

cs.DS↗
arXiv2017

Near Optimal Sketching of Low-Rank Tensor Regression

We study the least squares regression problem \begin{align*} \min_{Θ\in \mathcal{S}_{\odot D,R}} \|AΘ-b\|_2, \end{align*} where $\mathcal{S}_{\odot D,R}$ is the set of $Θ$ for which $Θ= \sum_{r=1}^{R} θ_1^{(r)} \circ \cdots \circ θ_D^{(r)}$ for vectors $θ_d^{(r)} \in \mathbb{R}^{p_d}$ for all $r \in [R]$ and $d \in [D]$, and $\circ$ denotes the outer product of vectors. That is, $Θ$ is a low-dimensional, low-rank tensor. This is motivated by the fact that the number of parameters in $Θ$ is only $R \cdot \sum_{d=1}^D p_d$, which is significantly smaller than the $\prod_{d=1}^{D} p_d$ number of parameters in ordinary least squares regression. We consider the above CP decomposition model of tensors $Θ$, as well as the Tucker decomposition. For both models we show how to apply data dimensionality reduction techniques based on {\it sparse} random projections $Φ\in \mathbb{R}^{m \times n}$, with $m \ll n$, to reduce the problem to a much smaller problem $\min_Θ \|ΦA Θ- Φb\|_2$, for which if $Θ'$ is a near-optimum to the smaller problem, then it is also a near optimum to the original problem. We obtain significantly smaller dimension and sparsity in $Φ$ than is possible for ordinary least squares regression, and we also provide a number of numerical simulations supporting our theory.

cs.LG↗
arXiv2017

Faster Kernel Ridge Regression Using Sketching and Preconditioning

Kernel Ridge Regression (KRR) is a simple yet powerful technique for non-parametric regression whose computation amounts to solving a linear system. This system is usually dense and highly ill-conditioned. In addition, the dimensions of the matrix are the same as the number of data points, so direct methods are unrealistic for large-scale datasets. In this paper, we propose a preconditioning technique for accelerating the solution of the aforementioned linear system. The preconditioner is based on random feature maps, such as random Fourier features, which have recently emerged as a powerful technique for speeding up and scaling the training of kernel-based methods, such as kernel ridge regression, by resorting to approximations. However, random feature maps only provide crude approximations to the kernel function, so delivering state-of-the-art results by directly solving the approximated system requires the number of random features to be very large. We show that random feature maps can be much more effective in forming preconditioners, since under certain conditions a not-too-large number of random features is sufficient to yield an effective preconditioner. We empirically evaluate our method and show it is highly effective for datasets of up to one million training examples.

math.NA↗
arXiv2017

Sharper Bounds for Regularized Data Fitting

We study matrix sketching methods for regularized variants of linear regression, low rank approximation, and canonical correlation analysis. Our main focus is on sketching techniques which preserve the objective function value for regularized problems, which is an area that has remained largely unexplored. We study regularization both in a fairly broad setting, and in the specific context of the popular and widely used technique of ridge regularization; for the latter, as applied to each of these problems, we show algorithmic resource bounds in which the {\em statistical dimension} appears in places where in previous bounds the rank would appear. The statistical dimension is always smaller than the rank, and decreases as the amount of regularization increases. In particular, for the ridge low-rank approximation problem $\min_{Y,X} \lVert YX - A \rVert_F^2 + λ\lVert Y\rVert_F^2 + λ\lVert X \rVert_F^2$, where $Y\in\mathbb{R}^{n\times k}$ and $X\in\mathbb{R}^{k\times d}$, we give an approximation algorithm needing \[ O(\mathtt{nnz}(A)) + \tilde{O}((n+d)\varepsilon^{-1}k \min\{k, \varepsilon^{-1}\mathtt{sd}_λ(Y^*)\})+ \mathtt{poly}(\mathtt{sd}_λ(Y^*) \varepsilon^{-1}) \] time, where $s_λ(Y^*)\le k$ is the statistical dimension of $Y^*$, $Y^*$ is an optimal $Y$, $\varepsilon$ is an error parameter, and $\mathtt{nnz}(A)$ is the number of nonzero entries of $A$.This is faster than prior work, even when $λ=0$. We also study regularization in a much more general setting. For example, we obtain sketching-based algorithms for the low-rank approximation problem $\min_{X,Y} \lVert YX - A \rVert_F^2 + f(Y,X)$ where $f(\cdot,\cdot)$ is a regularizing function satisfying some very general conditions (chiefly, invariance under orthogonal transformations).

cs.DS↗
arXiv2017

Fast Regression with an $\ell_\infty$ Guarantee

Sketching has emerged as a powerful technique for speeding up problems in numerical linear algebra, such as regression. In the overconstrained regression problem, one is given an $n \times d$ matrix $A$, with $n \gg d$, as well as an $n \times 1$ vector $b$, and one wants to find a vector $\hat{x}$ so as to minimize the residual error $\|Ax-b\|_2$. Using the sketch and solve paradigm, one first computes $S \cdot A$ and $S \cdot b$ for a randomly chosen matrix $S$, then outputs $x' = (SA)^{\dagger} Sb$ so as to minimize $\|SAx' - Sb\|_2$. The sketch-and-solve paradigm gives a bound on $\|x'-x^*\|_2$ when $A$ is well-conditioned. Our main result is that, when $S$ is the subsampled randomized Fourier/Hadamard transform, the error $x' - x^*$ behaves as if it lies in a "random" direction within this bound: for any fixed direction $a\in \mathbb{R}^d$, we have with $1 - d^{-c}$ probability that \[ \langle a, x'-x^*\rangle \lesssim \frac{\|a\|_2\|x'-x^*\|_2}{d^{\frac{1}{2}-γ}}, \quad (1) \] where $c, γ> 0$ are arbitrary constants. This implies $\|x'-x^*\|_{\infty}$ is a factor $d^{\frac{1}{2}-γ}$ smaller than $\|x'-x^*\|_2$. It also gives a better bound on the generalization of $x'$ to new examples: if rows of $A$ correspond to examples and columns to features, then our result gives a better bound for the error introduced by sketch-and-solve when classifying fresh examples. We show that not all oblivious subspace embeddings $S$ satisfy these properties. In particular, we give counterexamples showing that matrices based on Count-Sketch or leverage score sampling do not satisfy these properties. We also provide lower bounds, both on how small $\|x'-x^*\|_2$ can be, and for our new guarantee (1), showing that the subsampled randomized Fourier/Hadamard transform is nearly optimal.

cs.DS↗
arXiv2017

Algorithms for $\ell_p$ Low Rank Approximation

We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise $\ell_p$-approximation error, for any $p \geq 1$; the case $p = 2$ is the classical SVD problem. We obtain the first provably good approximation algorithms for this version of low-rank approximation that work for every value of $p \geq 1$, including $p = \infty$. Our algorithms are simple, easy to implement, work well in practice, and illustrate interesting tradeoffs between the approximation quality, the running time, and the rank of the approximating matrix.

cs.DS↗
arXiv2017

Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $Θ(\min\{n,\log(1/δ)\log^2(\frac n{\log(1/δ)})\})$ for failure probability $δ$. Our lower bound holds even if promised $\mathop{support}(y)\subset \mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $\ell_p$-sampling in strict turnstile streams for $0\le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised $x\in\{0,1\}^n$ at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm $\mathcal A$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $\mathcal A$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with $\mathcal A$, which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with $\mathcal A$ adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from $\mathcal A$ in the reduction.

cs.CC↗