Smoothed Analysis of Moore-Penrose Inversion

TL;DR

This paper conducts a smoothed analysis of the Moore-Penrose inverse's condition number, revealing that its expected value depends solely on matrix elongation, regardless of distribution specifics.

Contribution

It provides the first asymptotic analysis showing the expected condition number depends only on matrix elongation, not distribution parameters.

Findings

Expected condition number depends only on matrix elongation.

Asymptotic behavior is distribution-independent.

Results apply to rectangular matrices in smoothed analysis.

Abstract

We perform a smoothed analysis of the condition number of rectangular matrices. We prove that, asymptotically, the expected value of this condition number depends only of the elongation of the matrix, and not on the center and variance of the underlying probability distribution.