Adaptive Rank-One Matrix Completion Using Sum of Outer Products

Abstract

This paper presents a novel loss function referred to as hybrid ordinary-Welsch (HOW) and a new sparsity-inducing regularizer associated with HOW. We theoretically show that the regularizer is quasiconvex and that the corresponding Moreau envelope is convex. Moreover, the closed-form solution to its Moreau envelope, namely, the proximity operator, is derived. Compared with nonconvex regularizers like the ℓp-norm with 0 < p < 1 that requires iterations to find the corresponding proximity operator, the developed regularizer has a closed-form proximity operator. We apply our regularizer to the robust matrix completion problem, and develop an efficient algorithm based on the alternating direction method of multipliers. The convergence of the suggested method is analyzed and we prove that any generated accumulation point is a stationary point. Finally, experimental results based on synthetic and real-world datasets demonstrate that our algorithm is superior to the stateof-the-art methods in terms of restoration performance.

Robust Recommendation via Social Network Enhanced Matrix Completion

“…Matrix completion (MC) [1], [2] refers to recovering the missing entries of a partially-observed matrix. It has numerous applications in signal processing and machine learning, such as hyperspectral imaging [3] and image inpainting [4]. MC can be formulated as a constrained rank minimization problem [5], but it is NP-hard since the rank is discrete.…”