Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets
This work is significant for researchers in matrix sensing and recovery, as it provides theoretical guarantees for unique solutions under rank-one measurements, potentially simplifying recovery without explicit low-rank regularization.
This paper investigates the constraint set in low-rank, positive semidefinite (PSD) matrix sensing with rank-one measurements. It characterizes the radius of the set of PSD matrices satisfying the measurements, showing that a specific sampling rate guarantees singleton solution sets for exactly low-rank matrices.
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.