# SPARSE GRAPH BASED SKETCHING FOR FAST NUMERICAL LINEAR ALGEBRA

2021 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING (ICASSP 2021)（2021）

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 subset of R-n of dimension k, we show that the magical graph with left degree s = 2 yields a (1 +/- epsilon) l(2)-subspace embedding for U, if the number of right vertices (the sketch size) m = O(k(2)/epsilon(2)). The expander graph with s = O (log k/epsilon) yields a subspace embedding for m = 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.

MoreTranslated text

Key words

Randomized sketching, expander graphs, low-rank approximation, least squares regression

AI Read Science

Must-Reading Tree

Example

Generate MRT to find the research sequence of this paper

Chat Paper

Summary is being generated by the instructions you defined