Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints

A wide variety of applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the underlying problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical conditions have been derived to elucidate when this replacement is likely to be successful, these conditions are highly restrictive and convex algorithms fail when the ambient rank is too high or when the constraint set is poorly structured. Non-convex alternatives fare somewhat better when carefully tuned; however, convergence to locally optimal solutions remains a continuing source of failure. Under this backdrop we derive a deceptively simple, parameter-free Bayesian-inspired algorithm that is capable, over a wide battery of empirical tests, of successful recovery even approaching the theoretical limit where the number of measurements is equal to the degrees of freedom in the unknown low-rank matrix. Somewhat surprisingly, this is possible even when the affine constraint set is highly ill-conditioned. While proving general recovery conditions remains evasive for non-convex algorithms, Bayesian or otherwise, we nonetheless quantify special cases where our algorithm is guaranteed to succeed while existing algorithms may fail. We conclude with a simple computer vision application involving image rectification.