Proof methods for robust low-rank matrix recovery

Abstract

Low-rank matrix recovery problems arise naturally as mathematical formulations of various inverse problems, such as matrix completion, blind deconvolution, and phase retrieval. Over the last two decades, a number of works have rigorously analyzed the reconstruction performance for such scenarios, giving rise to a rather general understanding of the potential and the limitations of lowrank matrix models in sensing problems. In this article, we compare the two main proof techniques that have been paving the way to a rigorous analysis, discuss their potential and limitations, and survey their successful applications. On the one hand, we review approaches based on descent cone analysis, showing that they often lead to strong guarantees even in the presence of adversarial noise, but face limitations when it comes to structured observations. On the other hand, we discuss techniques using approximate dual certificates and the golfing scheme, which are often better suited to deal with practical measurement structures, but sometimes lead to weaker guarantees. Lastly, we review recent progress towards analyzing descent cones also for structured scenarios-exploiting the idea of splitting the cones into multiple parts that are analyzed via different techniques.

Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing

“…The nuclear norm minimization approach [12] has been studied for Matrix Completion in [13,14,15], for Phase Retrieval in [16,17,18], for Robust PCA in [19], and for Blind Deconvolution in [20] as well as its extension to the Blind Demixing problem in [21,22]. We refer also to the overview article [23] for further pointers to the literature. Several other approaches, which have been proposed in the literature, are the projected gradient method [24], the iterative greedy algorithm [25], and the Iteratively Reweighted Least Squares (IRLS) algorithm [26,27].…”

“…Suppose that eqs. (23) to (26) hold. Furthermore, suppose that both A and à satisfy the RIP with constant δ > 0.…”