Improved Algorithms for Low Rank Approximation from Sparsity

TL;DR

This paper introduces improved algorithms for low rank approximation assuming the low rank factors are sparse, significantly reducing computational time and memory usage for spectral norm and Frobenius norm errors.

Contribution

It presents novel algorithms that leverage sparsity of low rank factors to overcome longstanding computational and memory barriers in low rank approximation.

Findings

Reduced spectral norm approximation time to near-linear in non-zero entries

Achieved sublinear memory bounds in streaming models for Frobenius norm

Provided polynomial time algorithms with support on small submatrices

Abstract

We overcome two major bottlenecks in the study of low rank approximation by assuming the low rank factors themselves are sparse. Specifically, (1) for low rank approximation with spectral norm error, we show how to improve the best known $\mathsf{nnz}(\mathbf A) k / \sqrt{\varepsilon}$ running time to $\mathsf{nnz}(\mathbf A)/\sqrt{\varepsilon}$ running time plus low order terms depending on the sparsity of the low rank factors, and (2) for streaming algorithms for Frobenius norm error, we show how to bypass the known $\Omega(nk/\varepsilon)$ memory lower bound and obtain an $s k (\log n)/ \mathrm{poly}(\varepsilon)$ memory bound, where $s$ is the number of non-zeros of each low rank factor. Although this algorithm is inefficient, as it must be under standard complexity theoretic assumptions, we also present polynomial time algorithms using $\mathrm{poly}(s,k,\log n,\varepsilon^{-1})$ memory that output rank $k$ approximations supported on a $O(sk/\varepsilon)\times O(sk/\varepsilon)$ submatrix. Both the prior $\mathsf{nnz}(\mathbf A) k / \sqrt{\varepsilon}$ running time and the $nk/\varepsilon$ memory for these problems were long-standing barriers; our results give a natural way of overcoming them assuming sparsity of the low rank factors.