# Sensitivity of low-rank matrix recovery

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.

• 7 publications
• 11 publications
05/21/2017

### Improved Algorithms for Matrix Recovery from Rank-One Projections

We consider the problem of estimation of a low-rank matrix from a limite...
09/26/2019

### A theory of condition for unconstrained perturbations

Traditionally, the theory of condition numbers assumes errors in the dat...
08/04/2020

### A Schatten-q Matrix Perturbation Theory via Perturbation Projection Error Bound

This paper studies the Schatten-q error of low-rank matrix estimation by...
05/12/2022

### Sketching sparse low-rank matrices with near-optimal sample- and time-complexity

We consider the problem of recovering an n_1 × n_2 low-rank matrix with ...
03/14/2011

### Universal low-rank matrix recovery from Pauli measurements

We study the problem of reconstructing an unknown matrix M of rank r and...
09/04/2020

### Smoothed analysis of the condition number under low-rank perturbations

Let M be an arbitrary n by n matrix of rank n-k. We study the condition ...
05/24/2018

### Confidence region of singular vectors for high-dimensional and low-rank matrix regression

Let M∈R^m_1× m_2 be an unknown matrix with r= rank( M)≪(m_1,m_2) whose ...