Low-Rank Positive Semidefinite Matrix Recovery from Quadratic Measurements with Outliers
Year: 2016
Type: preprint
Abstract: We address the problem of estimating a low-rank positive semidefinite (PSD) matrix $\boldsymbol{X}\in\mathbb{R}^{n\times n}$ from a set of measurements $z_i= \boldsymbol{a}_i^T\boldsymbol{X}\boldsymbol{a}_i$, $i=1,\ldots, m$, which are quadratic in the sensing vectors $\boldsymbol{a}_i$'s composed of i.i.d. standard Gaussian entries. This problem arises from applications such as covariance sketching, quantum space tomography, and power spectrum estimation. Specifically, we consider a convex optimization problem that seeks the PSD matrix with minimum $\ell_1$ norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a rank-r PSD matrix $\boldsymbol{X}$ of size-n can be exactly recovered from $\mathcal{O}(nr^{2})$ measurements, even when a fraction of the measurements is corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of the rank of the PSD matrix, a non-convex algorithm based on subgradient descent is proposed that demonstrates superior empirical performance.
Source: arXiv (Cornell University)
Authors: Yuan‐Xin Li, Yue Sun, Yuejie Chi
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2261161581