  Although many techniques have been applied to matrix factorization (MF), they
may not fully exploit the feature structure. In this paper, we incorporate the
grouping effect into MF and propose a novel method called Robust Matrix
Factorization with Grouping effect (GRMF). The grouping effect is a
generalization of the sparsity effect, which conducts denoising by clustering
similar values around multiple centers instead of just around 0. Compared with
existing algorithms, the proposed GRMF can automatically learn the grouping
structure and sparsity in MF without prior knowledge, by introducing a
naturally adjustable non-convex regularization to achieve simultaneous sparsity
and grouping effect. Specifically, GRMF uses an efficient alternating
minimization framework to perform MF, in which the original non-convex problem
is first converted into a convex problem through Difference-of-Convex (DC)
programming, and then solved by Alternating Direction Method of Multipliers
(ADMM). In addition, GRMF can be easily extended to the Non-negative Matrix
Factorization (NMF) settings. Extensive experiments have been conducted using
real-world data sets with outliers and contaminated noise, where the
experimental results show that GRMF has promoted performance and robustness,
compared to five benchmark algorithms.
