A Level Set Method with Secant Iterations for the Least-Squares Constrained Nuclear Norm Minimization

TL;DR

This paper introduces an efficient algorithm combining level set and secant methods for nuclear norm minimization under least-squares constraints, with strong theoretical guarantees and superior numerical performance.

Contribution

It develops a novel algorithm that integrates a level set method with secant iterations and a proximal approach, with new theoretical insights on nonsingularity and convergence.

Findings

Algorithm outperforms existing methods in numerical tests.

Theoretical proof of nonsingularity of Clarke Jacobian.

Fast local convergence due to semismoothness.

Abstract

We present an efficient algorithm for least-squares constrained nuclear norm minimization, a computationally challenging problem with broad applications. Our approach combines a level set method with secant iterations and a proximal generation method. As a key theoretical contribution, we establish the nonsingularity of the Clarke generalized Jacobian for a general class of projection norm functions over closed convex sets. This property and the (strong) semismoothness of our value function yield fast local convergence of the secant method. For the resulting nuclear norm regularized subproblems, we develop a proximal generation method that exploits low-rank structures without compromising convergence. Extensive numerical experiments demonstrate the superior performance of our approach compared to state-of-the-art methods.