Sparse graph based sketching for fast numerical linear algebra

TL;DR

This paper investigates sparse graph-based sketching matrices, particularly from expander and magical graphs, for fast numerical linear algebra, demonstrating their theoretical embedding properties and practical efficiency.

Contribution

It introduces new theoretical results on subspace embeddings using sparse bipartite graphs and explores constructions with reduced randomness for efficient linear algebra computations.

Findings

Magical graphs with degree 2 achieve $(1 ext{±}\epsilon)$ subspace embedding with $m=O(k^2/\epsilon^2)$

Expander graphs with degree $O(rac{ ext{log} ext{k}}{ ext{ extepsilon}})$ achieve embeddings with $m=O(rac{k ext{log} ext{k}}{ ext{ extepsilon}^2})$

Sparse graph sketching matrices perform well in practice on synthetic and real datasets

Abstract

In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree~s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace $\mathcal{U} \subseteq \mathbb{R}^n$ of dimension $k$, we show that the magical graph with left degree $s=2$ yields a $(1\pm \epsilon)$ ${\ell}_2$-subspace embedding for $\mathcal{U}$, if the number of right vertices (the sketch size) $m = \mathcal{O}({k^2}/{\epsilon^2})$. The expander graph with $s = \mathcal{O}({\log k}/{\epsilon})$ yields a subspace embedding for $m = \mathcal{O}({k \log k}/{\epsilon^2})$. We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.