Approximation Algorithms for $\ell_0$-Low Rank Approximation

TL;DR

This paper introduces new approximation algorithms for the NP-hard $\,\ell_0$-Low Rank Approximation problem, achieving improved runtime and approximation guarantees, including the first provable algorithms for $k > 1$ and sublinear time solutions for $k=1$.

Contribution

The authors develop the first provable algorithms for $\,\ell_0$-Low Rank Approximation with $k > 1$, and provide sublinear time $(2+\epsilon)$-approximation for $k=1$, advancing the state of the art.

Findings

Poly$(mn)$ time algorithm for $k > 1$ with $O(k \log(n/k))$ rank and $O(k^2 \log(n/k))$ approximation.

Sublinear time $(2+\epsilon)$-approximation for $k=1$.

Improved approximation for binary matrices with $(1+O(\psi))$ factor.

Abstract

We study the $\ell_0$-Low Rank Approximation Problem, where the goal is, given an $m \times n$ matrix $A$, to output a rank-$k$ matrix $A'$ for which $\|A'-A\|_0$ is minimized. Here, for a matrix $B$, $\|B\|_0$ denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provide approximation algorithms which significantly improve the running time and approximation factor of previous work. For $k > 1$, we show how to find, in poly$(mn)$ time for every $k$, a rank $O(k \log(n/k))$ matrix $A'$ for which $\|A'-A\|_0 \leq O(k^2 \log(n/k)) \mathrm{OPT}$. To the best of our knowledge, this is the first algorithm with provable guarantees for the $\ell_0$-Low Rank Approximation Problem for $k > 1$, even for bicriteria algorithms. For the well-studied case when $k = 1$, we give a $(2+\epsilon)$-approximation in {\it sublinear time}, which is impossible for other variants of low rank approximation such as for the Frobenius norm. We strengthen this for the well-studied case of binary matrices to obtain a $(1+O(\psi))$-approximation in sublinear time, where $\psi = \mathrm{OPT}/\lVert A\rVert_0$. For small $\psi$, our approximation factor is $1+o(1)$.