Sparse Coresets for SVD on Infinite Streams

Abstract

In streaming Singular Value Decomposition (SVD), $d$-dimensional rows of a possibly infinite matrix arrive sequentially as points in ${\mathbb{R}}^d$. An $\epsilon$-coreset is a (much smaller) matrix whose sum of square distances of the rows to any hyperplane approximates that of the original matrix to a $1\pm \epsilon$ factor. Our main result is that we can maintain a $\epsilon$-coreset while storing only $O(d{\log }^{2}d/{\epsilon }^2)$ rows. Known lower bounds of $\Omega (d/{\epsilon }^2)$ rows show that this is nearly optimal. Moreover, each row of our coreset is a weighted subset of the input rows. This is highly desirable since it: (1) preserves sparsity; (2) is easily interpretable; (3) avoids precision errors; (4) applies to problems with constraints on the input. Previous streaming results for SVD that return a subset of the input required storing $\Omega (d{\log }^{3}n/{\epsilon }^2)$ rows where $n$ is the number of rows seen so far. Our algorithm, with storage independent of $n$, is the first result that uses finite memory on infinite streams. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.