New Subset Selection Algorithms for Low Rank Approximation: Offline and Online

TL;DR

This paper develops new algorithms for subset selection in low-rank matrix approximation, achieving near-optimal trade-offs for various loss functions in both offline and online settings, with significant theoretical improvements.

Contribution

It introduces nearly optimal offline bicriteria algorithms for entrywise losses and the first online algorithms for b5_pb5_p subspace and low-rank approximation, advancing the theoretical understanding.

Findings

Improved bicriteria algorithms for entrywise b5_pb5_p losses, tight for b5_1.

First oblivious b5_pb5_p subspace embeddings for 1<p<2 with nearly optimal distortion.

First online algorithms for b5_pb5_p subspace and low-rank approximation.

Abstract

Subset selection for the rank $k$ approximation of an $n\times d$ matrix $A$ offers improvements in the interpretability of matrices, as well as a variety of computational savings. This problem is well-understood when the error measure is the Frobenius norm, with various tight algorithms known even in challenging models such as the online model, where an algorithm must select the column subset irrevocably when the columns arrive one by one. In contrast, for other matrix losses, optimal trade-offs between the subset size and approximation quality have not been settled, even in the offline setting. We give a number of results towards closing these gaps. In the offline setting, we achieve nearly optimal bicriteria algorithms in two settings. First, we remove a $\sqrt k$ factor from a result of [SWZ19] when the loss function is any entrywise loss with an approximate triangle inequality and at least linear growth. Our result is tight for the $\ell_1$ loss. We give a similar improvement for entrywise $\ell_p$ losses for $p>2$, improving a previous distortion of $k^{1-1/p}$ to $k^{1/2-1/p}$. Our results come from a technique which replaces the use of a well-conditioned basis with a slightly larger spanning set for which any vector can be expressed as a linear combination with small Euclidean norm. We show that this technique also gives the first oblivious $\ell_p$ subspace embeddings for $1<p<2$ with $\tilde O(d^{1/p})$ distortion, which is nearly optimal and closes a long line of work. In the online setting, we give the first online subset selection algorithm for $\ell_p$ subspace approximation and entrywise $\ell_p$ low rank approximation by implementing sensitivity sampling online, which is challenging due to the sequential nature of sensitivity sampling. Our main technique is an online algorithm for detecting when an approximately optimal subspace changes substantially.