A Novel Nonconvex Relaxation Approach to Low-Rank Matrix Completion of Inexact Observed Data

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2378-2410, September 2024.
Abstract. In recent years, matrix completion has become one of the main concepts in data science. In the process of data acquisition in real applications, in addition to missing data, observed data may be inaccurate. This paper is concerned with such matrix completion of inexact observed data which can be modeled as a rank minimization problem. We adopt the difference of the nuclear norm and the Frobenius norm as an approximation of the rank function, employ Tikhonov-type regularization to preserve the inherent characteristics of original data and control oscillation arising from inexact observations, and then establish a new nonsmooth and nonconvex relaxation model for such low-rank matrix completion. We propose a new accelerated proximal gradient–type algorithm to solve the nonsmooth and nonconvex minimization problem and show that the generated sequence is bounded and globally converges to a critical point of our model. Furthermore, the rate of convergence is given via the Kurdyka–Łojasiewicz property. We evaluate our model and method on visual images and received signal strength fingerprint data in an indoor positioning system. Numerical experiments illustrate that our approach outperforms some state-of-the-art methods, and also verify the efficacy of the Tikhonov-type regularization.