Statistical Rounding Error Analysis for Random Matrix Computations

TL;DR

This paper introduces a statistical approach to analyze rounding errors in random matrix computations, providing more accurate bounds than traditional worst-case analysis, with applications in wireless communications, signal processing, and machine learning.

Contribution

It develops a new statistical rounding error analysis assuming independent errors, deriving closed-form expressions for expectation and variance in random matrix computations.

Findings

Analytical expressions are at least two orders of magnitude tighter than worst-case bounds.

Numerical experiments validate the accuracy of the derived formulas.

The approach applies to key computations like inner products in random matrices.

Abstract

The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for random matrix computations. Such computations have numerous applications in the field of wireless communications, signal processing, and machine learning. By assuming the relative errors are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computations for random matrices. Numerical experiments validate the accuracy of our derivations and demonstrate that our analytical expressions are generally at least two orders of magnitude tighter than alternative worst-case bounds, exemplified through the inner products.