Truncated quadratic norm minimization for bilinear factorization based matrix completion

Abstract

Low-rank matrix completion is an important research topic with a wide range of applications. One prevailing way for matrix recovery is based on rank minimization. Directly solving this problem is NP hard. Therefore, various rank surrogates are developed, like nuclear norm. However, nuclear norm regularization minimizes the sum of all the singular values, and hence the rank is not well approximated. We propose a new rank substitution named truncated quadratic norm that performs the corresponding truncated quadratic operation on the singular values. This function takes the square of the minor singular values and maps large singular values to one. In order to reduce computational complexity, the original target matrix is factorized into two small matrices on which the truncated quadratic norm is imposed. The resultant problem is then solved by alternating minimization. We also prove that the solution sequence is able to converge to a critical point. Experimental results on synthetic data and real-world images demonstrate the excellent performance of our method in terms of recovery accuracy.