Approximation of rank function and its application to the nearest low-rank correlation matrix

Authors:
Shujun Bi;Le Han;Shaohua Pan
Affiliations:
Department of Mathematics, South China University of Technology, Tianhe District, Guangzhou, China;Department of Mathematics, South China University of Technology, Tianhe District, Guangzhou, China;Department of Mathematics, South China University of Technology, Tianhe District, Guangzhou, China
Venue:
Journal of Global Optimization
Year:
2013

Citing 21
Cited 0

Shape and motion from image streams under orthography: a factorization method

International Journal of Computer Vision
.879-approximation algorithms for MAX CUT and MAX 2SAT

STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
A class of smoothing functions for nonlinear and mixed complementarity problems

Computational Optimization and Applications
Derivatives of spectral functions

Mathematics of Operations Research
A Sequential Factorization Method for Recovering Shape and Motion From Image Streams

IEEE Transactions on Pattern Analysis and Machine Intelligence
Strong Semismoothness of Eigenvalues of Symmetric Matrices and Its Application to Inverse Eigenvalue Problems

SIAM Journal on Numerical Analysis
Semismooth Matrix-Valued Functions

Mathematics of Operations Research
A Dual Approach to Semidefinite Least-Squares Problems

SIAM Journal on Matrix Analysis and Applications
Fast maximum margin matrix factorization for collaborative prediction

ICML '05 Proceedings of the 22nd international conference on Machine learning
Least-Squares Covariance Matrix Adjustment

SIAM Journal on Matrix Analysis and Applications
A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

SIAM Journal on Matrix Analysis and Applications
The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications

Mathematics of Operations Research
An inexact primal–dual path following algorithm for convex quadratic SDP

Mathematical Programming: Series A and B
Matrix Methods in Data Mining and Pattern Recognition (Fundamentals of Algorithms)

Matrix Methods in Data Mining and Pattern Recognition (Fundamentals of Algorithms)
Exact Matrix Completion via Convex Optimization

Foundations of Computational Mathematics
Interior-Point Method for Nuclear Norm Approximation with Application to System Identification

SIAM Journal on Matrix Analysis and Applications
A Singular Value Thresholding Algorithm for Matrix Completion

SIAM Journal on Optimization
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

SIAM Review
Fixed point and Bregman iterative methods for matrix rank minimization

Mathematical Programming: Series A and B
An implementable proximal point algorithmic framework for nuclear norm minimization

Mathematical Programming: Series A and B
A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem

SIAM Journal on Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641---1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.