Comparison of Matrix Norm Sparsification

TL;DR

This paper explores how sparsification guarantees in different Schatten p-norms relate for matrices, revealing a surprising negative result for most p and q values, contrasting with vector cases.

Contribution

It provides the first analysis of the relationship between matrix sparsification in different Schatten p-norms, showing most p and q pairs do not imply each other.

Findings

Negative results for most p and q pairs in matrix Schatten norms

Contrasts with positive vector Schatten norm sparsification results

Explicit constructions of matrices demonstrating these properties

Abstract

A well-known approach in the design of efficient algorithms, called matrix sparsification, approximates a matrix $A$ with a sparse matrix $A'$. Achlioptas and McSherry [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that $\|A'-A\|\leq \epsilon \|A\|$ for error parameter $\epsilon>0$. Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten $p$-norm for general $p$, which includes the spectral norm as the special case $p=\infty$. We investigate the relation between fixed but different $p\neq q$, that is, whether sparsification in the Schatten $p$-norm implies (existentially and/or algorithmically) sparsification in the Schatten $q\text{-norm}$ with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of $p$ to focus on. Our main finding is a surprising contrast between this question and the analogous case of $\ell_p$-norm sparsification for vectors: For vectors, the answer is affirmative for $p<q$ and negative for $p>q$, but for matrices we answer negatively for almost all sufficiently distinct $p\neq q$. In addition, our explicit constructions may be of independent interest.