Analysis of randomized CholeskyQR for sparse matrices

TL;DR

This paper analyzes the rounding errors of randomized CholeskyQR algorithms for sparse matrices, introducing a new sparsity model and comparing theoretical and experimental bounds, revealing new phenomena in sparse cases.

Contribution

It introduces a new sparsity model and provides a detailed rounding error analysis for randomized CholeskyQR algorithms applied to sparse matrices.

Findings

New sparsity model for sparse matrices

Comparison of bounds with different models both theoretically and experimentally

Identification of new phenomena in randomized CholeskyQR for sparse matrices

Abstract

This work is about rounding error analysis of randomized CholeskyQR-type algorithms for sparse matrices. We often encounter QR factorization of the sparse matrices in many real problems. In this work, we focus on some typical CholeskyQR-type algorithms with matrix sketching, which is a popular randomized technique in recent years. We build a new model of the sparse matrices and provide rounding error analysis of randomized CholeskyQR-type algorithms for the sparse cases with this model. We make comparison between the bounds with different models of sparsity both theoretically and experimentally. Numerical experiments show some new phenomena of randomized CholeskyQR-type algorithms for the sparse cases, which do not occur in the common sparse cases. We also test the applicability, accuracy, efficiency and robustness of randomized CholeskyQR-type algorithms for sparse matrices.