Publication
ICASSP 2021
Conference paper

Sparse graph based sketching for fast numerical linear algebra

View publication

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 U ⊆ Rn of dimension k, we show that the magical graph with left degree s = 2 yields a (1 ± ε) 2-subspace embedding for U, if the number of right vertices (the sketch size) m = O(k2/ε2). The expander graph with s = O(log k/ε) yields a subspace embedding for m = O(k log k/ε2). 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.

Date

06 Jun 2021

Publication

ICASSP 2021

Authors

Tags

Share