Robust nonnegative matrix factorization with ordered structure constraints

Abstract

Abstract-Nonnegative matrix factorization (NMF) as a popular technique to find parts-based representations of nonnegative data has been widely used in real-world applications. Often the data which these applications process, such as motion sequences and video clips, are with ordered structure, i.e., consecutive neighbouring data samples are very likely share similar features unless a sudden change occurs. Therefore, traditional NMF assumes the data samples and features to be independently distributed, making it not proper for the analysis of such data. In this paper, we propose an ordered robust NMF (ORNMF) by capturing the embedded ordered structure to improve the accuracy of data representation. With a novel neighbour penalty term, ORNMF enforces the similarity of neighbouring data. ORNMF also adopts the L2,1-norm based loss function to improve its robustness against noises and outliers. A new iterative updating optimization algorithm is derived to solve ORNMF's objective function. The proofs of the convergence and correctness of the scheme are also presented. Experiments on both synthetic and real-world datasets have demonstrated the effectiveness of ORNMF.

Multiview Clustering via Robust Neighboring Constraint Nonnegative Matrix Factorization

“…A structured LRR was proposed by factorizing into the latent low-dimensional data-cluster representations, which characterize the data clustering structure for each view [1]. Meanwhile, NMF-based methods [19] were also proved to be useful, which enforce the constraint that the elements of the factor matrices must be nonnegative. It shows that when the Frobenius norm is used as a divergence, NMF is equivalent to a relaxed form of K-means clustering method.…”

“…In MVC, to learn the neighbor information in each view, based on ORNMF [19] which is a robust representation approach, the proposed rNNMF can be expressed as min…”

“…For each view, the updating rule of H (v) and W (v) satisfies the theorem in [24], which guarantees the correctness of the rule. e correctness analysis and convergence proof of (10) and (12) are shown based on the method [19] in the following section.…”

“…We now prove the convergence of the updating rule (10) using the auxiliary function approach in [19]. e definition of the auxiliary function is as follows.…”