Matrix Completion and Related Problems via Strong Duality

TL;DR

This paper establishes a strong duality framework for non-convex matrix factorization problems, enabling global optimality analysis and applying it to matrix completion and robust PCA with near-optimal guarantees.

Contribution

It introduces a novel analytical framework for strong duality in non-convex matrix factorization, linking primal and dual solutions and enabling convex reformulations.

Findings

Strong duality holds for certain non-convex matrix problems.

Exact recovery is achievable with near-optimal sample complexity.

Framework applies to matrix completion and robust PCA.

Abstract

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.