Guaranteed Minimum Rank Approximation from Linear Observations by Nuclear Norm Minimization with an Ellipsoidal Constraint

TL;DR

This paper provides explicit performance guarantees for nuclear norm minimization with an ellipsoidal constraint, ensuring reliable low-rank matrix approximation from linear observations even with noise.

Contribution

It introduces a theoretical performance bound for nuclear norm minimization under ellipsoidal constraints, extending previous results to noisy and approximate low-rank scenarios.

Findings

Derived explicit error bounds for nuclear norm minimization with ellipsoidal constraints

Extended theoretical guarantees to noisy measurement settings

Validated polynomial-time algorithms for approximate low-rank matrix recovery

Abstract

The rank minimization problem is to find the lowest-rank matrix in a given set. Nuclear norm minimization has been proposed as an convex relaxation of rank minimization. Recht, Fazel, and Parrilo have shown that nuclear norm minimization subject to an affine constraint is equivalent to rank minimization under a certain condition given in terms of the rank-restricted isometry property. However, in the presence of measurement noise, or with only approximately low rank generative model, the appropriate constraint set is an ellipsoid rather than an affine space. There exist polynomial-time algorithms to solve the nuclear norm minimization with an ellipsoidal constraint, but no performance guarantee has been shown for these algorithms. In this paper, we derive such an explicit performance guarantee, bounding the error in the approximate solution provided by nuclear norm minimization with an ellipsoidal constraint.