A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable if there are at most finitely many rank-$r$ matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a deterministic sampling condition for finite completability. We use this to also derive deterministic sampling conditions for unique completability that can be efficiently verified. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if $O(\max\{r,\log d\})$ entries per column are observed. These findings have several implications on LRMC regarding lower bounds, sample and computational complexity, the role of coherence, adaptive settings and the validation of any completion algorithm. We complement our theoretical results with experiments that support our findings and motivate future analysis of uncharted sampling regimes.Comment: This update corrects an error in version 2 of this paper, where we erroneously assumed that columns with more than r+1 observed entries would yield multiple independent constraint

A Deterministic Theory for Exact Non-Convex Phase Retrieval

In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the convergence of the WF iterates to a true solution is attained geometrically under a single condition on the lifted forward model. As a result, a deterministic relationship between the accuracy of spectral initialization and the validity of {the regularity condition} is derived. In particular, we determine that a certain concentration property on the spectral matrix must hold uniformly with a sufficiently tight constant. This culminates into a sufficient condition that is equivalent to a restricted isometry-type property over rank-1, positive semi-definite matrices, and amounts to a less stringent requirement on the lifted forward model than those of prominent low-rank-matrix-recovery methods in the literature. We characterize the performance limits of our framework in terms of the tightness of the concentration property via novel bounds on the convergence rate and on the signal-to-noise ratio such that the theoretical guarantees are valid using the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin