Low-rank matrix completion in a general non-orthogonal basis

Abstract

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach. We introduce a condition on the basis called the correlation condition. This condition can be computed in time O(n3) and holds for many cases of deterministic basis. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter ν, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from O(nrνlog2⁡n) uniformly random expansion coefficients.