A Matrix Factorization Approach in Turnstile Streaming

Abstract

We define the $M$-point query problem in data streams. Given a fixed matrix $M$, the goal is to maintain a vector $x$ under turnstile updates and answer each query $u$ with an estimate ${\hat{y}}_u$ satisfying $|y_u-{\hat{y}}_u|\le \epsilon \Vert x{\Vert }_1$, where $y=Mx$. We show that if $M$ admits a factorization $M=AB$, where $A$ and $B$ have space-efficient representations, then there is a streaming algorithm using $O({\epsilon }^{-1}\Vert A{\Vert }_{2\to \infty }\Vert B{\Vert }_{1\to 1}+({\epsilon }^{-1}\Vert A{\Vert }_{\infty \to \infty }\Vert B{\Vert }_{1\to 1}{)}^{2/3})$ words of memory. An important special case is the lower-triangular all-ones matrix, which corresponds to the quantiles problem with additive error $\pm \epsilon n$, where $n$ is the database size. Our framework generalizes the dyadic approach of Cormode and Muthukrishnan (J. Algorithms, 2005) for turnstile quantiles, and simplifies and improves the analysis of the state-of-the-art dyadic CountSketch algorithms of Wang et al. (SIGMOD, 2013) and Luo et al. (VLDB, 2016). Our approach is also related to the matrix mechanism of Li et al. (VLDB J., 2015) in differential privacy: given a database $x\in {\mathbb{R}}^U$ and a matrix $M$, the mechanism outputs a private approximation to $Mx$, with the privacy-error tradeoff governed by a matrix factorization norm of $M$. We also improve the prior lower bound for quantiles with deletions, showing a memory lower bound of $\hat{I}\text{c◯}({\epsilon }^{-1}\log U)$ words. We also show any factorization has $\Vert A{\Vert }_{2\to \infty }\Vert B{\Vert }_{1\to 1}=\hat{I}\text{c◯}(({\log }^{1.5}U)/\log \log U)$. This lower bound is new, and shows that for quantiles, the dyadic CountSketch is nearly optimal amongst factorization-based approaches.