Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Chiyu Ma, Jiaming Ma, Defeng Sun

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.

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

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.
Paper Structure (13 sections, 15 theorems, 59 equations, 7 tables, 3 algorithms)

This paper contains 13 sections, 15 theorems, 59 equations, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A level set method with secant iterations for \ref{['eq:lscnnm']}
  4. A level set method with the value function $\varphi(\lambda)$
  5. Nonsingularity of the Clarke generalized Jacobian $\partial \varphi(\lambda)$
  6. A secant method for the univariate equation \ref{['eq:une']}
  7. A proximal generation method for \ref{['eq:nnrls']}
  8. Numerical experiments
  9. Numerical Issues in Algorithm \ref{['alg:rsls-sec']}
  10. ADMM algorithms for \ref{['eq:lscnnm']}
  11. Matrix regression problems
  12. Matrix completion problems
  13. Conclusion and future work

Key Result

Proposition 1

Let $F:\mathcal{O}\subset\mathcal{X}\to\mathcal{Y}$ be a locally Lipschitz continuous function on the open set $\mathcal{O}$ and $x\in\mathcal{O}$. The following statements are equivalent for any $\gamma>0$: (i) $F$ is $\gamma-$order semismooth at $x$, i.e., eq:def-gammassm and eq:cbd hold; (ii) for (iii) for any $h\to 0$ and $x+h\in D_F$, it holds

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • Proposition 3
  • proof
  • ...and 25 more