Restricted strong convexity and weighted matrix completion: optimal bounds with noise

Authors:
Sahand Negahban;Martin J. Wainwright
Affiliations:
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA;Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA and Department of Statistics
Venue:
The Journal of Machine Learning Research
Year:
2012

Citing 13
Cited 0

Matrix analysis

Matrix analysis
Learning with matrix factorizations

Learning with matrix factorizations
Consistency of Trace Norm Minimization

The Journal of Machine Learning Research
Exact Matrix Completion via Convex Optimization

Foundations of Computational Mathematics
Geometry of Cuts and Metrics

Geometry of Cuts and Metrics
The power of convex relaxation: near-optimal matrix completion

IEEE Transactions on Information Theory
Matrix completion from a few entries

IEEE Transactions on Information Theory
Matrix Completion from Noisy Entries

The Journal of Machine Learning Research
Spectral Regularization Algorithms for Learning Large Incomplete Matrices

The Journal of Machine Learning Research
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

SIAM Review
A Simpler Approach to Matrix Completion

The Journal of Machine Learning Research
Strong converse for identification via quantum channels

IEEE Transactions on Information Theory
Recovering Low-Rank Matrices From Few Coefficients in Any Basis

IEEE Transactions on Information Theory

Quantified Score

Hi-index	0.00

Visualization

Abstract

We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniformentrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the 'spikiness' and 'low-rankness' of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an M-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with lq-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.