Nonlinear matrix recovery using optimization on the Grassmann manifold

TL;DR

This paper introduces novel algorithms based on Riemannian optimization and alternating minimization for recovering high-rank matrices with nonlinear structures, providing theoretical guarantees and demonstrating high accuracy in numerical experiments.

Contribution

It develops new non-convex optimization algorithms on the Grassmann manifold for nonlinear matrix recovery, with proven convergence and complexity bounds.

Findings

Algorithms achieve high accuracy in matrix recovery.

Theoretical guarantees include global convergence and complexity bounds.

Methods are competitive with existing approaches.

Abstract

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.