Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets

TL;DR

This paper analyzes how rank-one measurement constraints influence the uniqueness of solutions in low-rank PSD matrix sensing, showing that under certain sampling rates, the solution set is a singleton regardless of the algorithm used.

Contribution

It characterizes the feasible set size in low-rank PSD matrix sensing with rank-one measurements and establishes sampling conditions for unique recovery independent of the chosen method.

Findings

Sampling rate guarantees singleton solution sets for exact low-rank matrices.

Feasible set radius can be characterized based on measurement properties.

Practical implications for PSD matrix recovery without low-rank regularization.

Abstract

We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.