Sensitivity of low-rank matrix recovery

TL;DR

This paper analyzes the sensitivity of low-rank matrix recovery from linear measurements, providing formulas and algorithms to compute condition numbers, and demonstrating how measurement quantity influences stability.

Contribution

It introduces an explicit formula for the condition number of low-rank approximation, linking it to singular value gaps, and presents an algorithm to compute this condition number.

Findings

Condition number depends on singular value gaps.

Number of measurements affects recovery stability.

Algorithm for computing condition number is demonstrated.

Abstract

We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of linear measurements affects it. In addition, we study the condition number of the rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for the condition number, which shows that it does depend on the relative singular value gap between the rth and (r+1)th singular values of the input matrix.