Adaptive Graph via Multiple Kernel Learning for Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) has been continuously evolving in several areas like pattern recognition and information retrieval methods. It factorizes a matrix into a product of 2 low-rank non-negative matrices that will define parts-based, and linear representation of nonnegative data. Recently, Graph regularized NMF (GrNMF) is proposed to find a compact representation,which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure. In GNMF, an affinity graph is constructed from the original data space to encode the geometrical information. In this paper, we propose a novel idea which engages a Multiple Kernel Learning approach into refining the graph structure that reflects the factorization of the matrix and the new data space. The GrNMF is improved by utilizing the graph refined by the kernel learning, and then a novel kernel learning method is introduced under the GrNMF framework. Our approach shows encouraging results of the proposed algorithm in comparison to the state-of-the-art clustering algorithms like NMF, GrNMF, SVD etc.Comment: This paper has been withdrawn by the author due to the terrible writin

The Diagonalized Newton Algorithm for Nonnegative Matrix Factorization

Non-negative matrix factorization (NMF) has become a popular machine learning approach to many problems in text mining, speech and image processing, bio-informatics and seismic data analysis to name a few. In NMF, a matrix of non-negative data is approximated by the low-rank product of two matrices with non-negative entries. In this paper, the approximation quality is measured by the Kullback-Leibler divergence between the data and its low-rank reconstruction. The existence of the simple multiplicative update (MU) algorithm for computing the matrix factors has contributed to the success of NMF. Despite the availability of algorithms showing faster convergence, MU remains popular due to its simplicity. In this paper, a diagonalized Newton algorithm (DNA) is proposed showing faster convergence while the implementation remains simple and suitable for high-rank problems. The DNA algorithm is applied to various publicly available data sets, showing a substantial speed-up on modern hardware.Comment: 8 pages + references; International Conference on Learning Representations, 201

Using Underapproximations for Sparse Nonnegative Matrix Factorization

Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g., text mining, image processing, microarray data analysis, collaborative filtering, etc. We introduce a novel approach to solve NMF problems, based on the use of an underapproximation technique, and show its effectiveness to obtain sparse solutions. This approach, based on Lagrangian relaxation, allows the resolution of NMF problems in a recursive fashion. We also prove that the underapproximation problem is NP-hard for any fixed factorization rank, using a reduction of the maximum edge biclique problem in bipartite graphs. We test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part-based representations with low reconstruction error. Our results are comparable and sometimes superior to those obtained by two standard Sparse Nonnegative Matrix Factorization techniques.Comment: Version 2 removed the section about convex reformulations, which was not central to the development of our main results; added material to the introduction; added a review of previous related work (section 2.3); completely rewritten the last part (section 4) to provide extensive numerical results supporting our claims. Accepted in J. of Pattern Recognitio