Optimal Sampling Algorithms for Block Matrix Multiplication

TL;DR

This paper develops optimized randomized sampling algorithms for block matrix multiplication, deriving optimal sampling probabilities and block sizes, and demonstrating superior performance through numerical experiments.

Contribution

It introduces new sampling algorithms based on A-optimal design, including practical modifications and a two-step method, advancing the efficiency of block matrix multiplication.

Findings

Algorithms outperform existing methods in numerical tests.

Optimal sampling probabilities derived from A-optimal design.

Modified block sizes reduce computational costs.

Abstract

In this paper, we investigate the randomized algorithms for block matrix multiplication from random sampling perspective. Based on the A-optimal design criterion, the optimal sampling probabilities and sampling block sizes are obtained. To improve the practicability of the block sizes, two modified ones with less computation cost are provided. With respect to the second one, a two step algorithm is also devised. Moreover, the probability error bounds for the proposed algorithms are given. Extensive numerical results show that our methods outperform the existing one in the literature.