Unified View of Matrix Completion under General Structural Constraints

Abstract

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.

published in NIPS 2015. Advances in Neural Information Processing Systems 28, 2015

References in corpus (6)

• Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
• Clustered Multi-Task Learning: A Convex Formulation
• Estimation with Norm Regularization
• Exponential Family Matrix Completion under Structural Constraints
• Estimation in high dimensions: a geometric perspective
• Sparse Prediction with the $k$-Support Norm

Cited by in corpus (3)

• Bypassing the Ambient Dimension: Private SGD with Gradient Subspace Identification
• Low-rank matrix completion and denoising under Poisson noise
• Convergence for nonconvex ADMM, with applications to CT imaging