Generalized Nonconvex Nonsmooth Low-Rank Minimization

04/29/2014
by   Canyi Lu, et al.
0

As surrogate functions of L_0-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on [0,∞). Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.

READ FULL TEXT

page 7

page 8

research
12/06/2014

Generalized Singular Value Thresholding

This work studies the Generalized Singular Value Thresholding (GSVT) ope...
research
01/29/2014

Smoothed Low Rank and Sparse Matrix Recovery by Iteratively Reweighted Least Squares Minimization

This work presents a general framework for solving the low rank and/or s...
research
07/02/2018

A nonconvex approach to low-rank and sparse matrix decomposition with application to video surveillance

In this paper, we develop a new nonconvex approach to the problem of low...
research
04/29/2016

Improved Sparse Low-Rank Matrix Estimation

We address the problem of estimating a sparse low-rank matrix from its n...
research
12/06/2017

A High-resolution DOA Estimation Method with a Family of Nonconvex Penalties

The low-rank matrix reconstruction (LRMR) approach is widely used in dir...
research
11/03/2018

Biconvex Landscape In SDP-Related Learning

Many machine learning problems can be reduced to learning a low-rank pos...
research
06/10/2016

Extended Gauss-Newton and Gauss-Newton-ADMM Algorithms for Low-Rank Matrix Optimization

We develop a generic Gauss-Newton (GN) framework for solving a class of ...

Please sign up or login with your details

Forgot password? Click here to reset