Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing

2026-08-23

AI Summary

本文研究了低秩半正定矩阵感知映射的稳定性问题，通过引入双Lipschitz常数并定义全局条件数，给出了确定性下界，并证明了随机秩一高斯测量在渐近意义下的最优性。

Abstract

In this paper we focus on the stability of positive semidefinite matrix sensing maps $Φ_{\mathcal{A}}(X)=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $Φ_{\mathcal{A}}(X)$ and define the global condition numbers as the ratio of upper and lower Lipschitz constants. We give deterministic universal lower bounds for these condition numbers, that depend only on the rank $r$ and on the underlying field. We then investigate the random rank-one Gaussian measurements and show that our lower bounds on condition numbers are asymptotically sharp, and therefore the random rank-one Gaussian measurements are asymptotically optimal. As an application, we derive the stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale.

Problem

• low-rank positive semidefinite matrix
• sensing maps
• bi-Lipschitz constants
• global condition numbers
• random rank-one Gaussian measurements

Innovation

• bi-Lipschitz constants
• global condition numbers
• rank-one Gaussian measurements
• stability guarantees