Optimal exact least squares rank minimization

Authors:
Shuo Xiang;Yunzhang Zhu;Xiaotong Shen;Jieping Ye
Affiliations:
Arizona State University, Tempe, AZ, USA;University of Minnesota, Minneapolis, MN, USA;University of Minnesota, Minneapolis, MN, USA;Arizona State University, Tempe, AZ, USA
Venue:
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Year:
2012

Citing 10
Cited 1

Sparse Approximate Solutions to Linear Systems

SIAM Journal on Computing
Uncovering shared structures in multiclass classification

Proceedings of the 24th international conference on Machine learning
Generalized Rank-Constrained Matrix Approximations

SIAM Journal on Matrix Analysis and Applications
An accelerated gradient method for trace norm minimization

ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
Exact Matrix Completion via Convex Optimization

Foundations of Computational Mathematics
Learning incoherent sparse and low-rank patterns from multiple tasks

Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining
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
Trace Norm Regularization: Reformulations, Algorithms, and Multi-Task Learning

SIAM Journal on Optimization
Low-rank estimation of higher order statistics

IEEE Transactions on Signal Processing

On the equivalent of low-rank linear regressions and linear discriminant analysis based regressions

Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining

Quantified Score

Hi-index	0.00

Visualization

Abstract

In multivariate analysis, rank minimization emerges when a low-rank structure of matrices is desired as well as a small estimation error. Rank minimization is nonconvex and generally NP-hard, imposing one major challenge. In this paper, we consider a nonconvex least squares formulation, which seeks to minimize the least squares loss function with the rank constraint. Computationally, we develop efficient algorithms to compute a global solution as well as an entire regularization solution path. Theoretically, we show that our method reconstructs the oracle estimator exactly from noisy data. As a result, it recovers the true rank optimally against any method and leads to sharper parameter estimation over its counterpart. Finally, the utility of the proposed method is demonstrated by simulations and image reconstruction from noisy background.