Date of this Version
Electronic Journal of Statistics
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.
The original and published work is available at: https://projecteuclid.org/euclid.ejs/1464710239#info
Compressed sensing, low-rank matrix, matrix completion, max-norm constrained minimization, minimax optimality, nonuniform sampling, sparsity
Cai, T., & Zhou, W. (2016). Matrix Completion via Max-Norm Constrained Optimization. Electronic Journal of Statistics, 10 (1), 1493-1525. http://dx.doi.org/10.1214/16-EJS1147
Date Posted: 27 November 2017
This document has been peer reviewed.