Unified View of Matrix Completion under General Structural Constraints
This work offers a theoretical framework for structured matrix completion, potentially benefiting researchers in machine learning and statistics, but it appears incremental as it extends existing analysis to more general norms.
The paper tackles the problem of matrix completion under general low-dimensional structural constraints by providing a unified analysis of sample complexity and estimation error for two estimators, establishing intermediate results on complexity measures and restricted strong convexity.
In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by {\em any} norm regularization. We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimation error. Our analysis relies on results from generic chaining, and we establish two intermediate results of independent interest: (a) in characterizing the size or complexity of low dimensional subsets in high dimensional ambient space, a certain partial complexity measure encountered in the analysis of matrix completion problems is characterized in terms of a well understood complexity measure of Gaussian widths, and (b) it is shown that a form of restricted strong convexity holds for matrix completion problems under general norm regularization. Further, we provide several non-trivial examples of structures included in our framework, notably the recently proposed spectral $k$-support norm.