Algorithms for $\ell_p$ Low Rank Approximation

TL;DR

This paper introduces new algorithms for low-rank matrix approximation minimizing entrywise _p errors for all p , including the classical case p=2 and the max norm p=0, with practical efficiency and theoretical guarantees.

Contribution

It provides the first provably effective algorithms for _p low-rank approximation applicable to all p , including the max norm, with simple implementation and practical performance.

Findings

Algorithms work well in practice.

Tradeoffs between approximation quality, runtime, and rank.

First provably good algorithms for all p .

Abstract

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.