AI 中文总结
该研究针对低秩半正定矩阵感知映射，定义全局条件数并给出其确定性通用下界，证明随机秩一高斯测量可达到该下界实现渐近最优，还推导了$\u03b9_1$-残差PhaseLift型估计器的稳定性保证。
AI 中文摘要
本文聚焦半正定矩阵感知映射$\u03a6_{\mathcal{A}}(X)=(\langle A_i,X\rangle)_{i=1}^m$的稳定性，其中$A_i\succeq0$、$X\succeq0$且$\u0072\u0061\u006e\u006b(X)\le r$。我们引入$\u03a6_{\mathcal{A}}(X)$的双利普希茨常数，并将全局条件数定义为上、下利普希茨常数的比值。我们给出这些条件数的确定性通用下界，其仅依赖于秩$r$和基础域。随后研究随机秩一高斯测量，证明我们的条件数下界是渐近尖锐的，因此随机秩一高斯测量是渐近最优的。作为应用，我们在最优采样规模下推导$\u03b9_1$-残差PhaseLift型估计器的稳定性保证。
英文摘要
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.
