Sequential Dimensionality Reduction for Extracting Localized Features

Linear dimensionality reduction techniques are powerful tools for image analysis as they allow the identification of important features in a data set. In particular, nonnegative matrix factorization (NMF) has become very popular as it is able to extract sparse, localized and easily interpretable features by imposing an additive combination of nonnegative basis elements. Nonnegative matrix underapproximation (NMU) is a closely related technique that has the advantage to identify features sequentially. In this paper, we propose a variant of NMU that is particularly well suited for image analysis as it incorporates the spatial information, that is, it takes into account the fact that neighboring pixels are more likely to be contained in the same features, and favors the extraction of localized features by looking for sparse basis elements. We show that our new approach competes favorably with comparable state-of-the-art techniques on synthetic, facial and hyperspectral image data sets.Comment: 24 pages, 12 figures. New numerical experiments on synthetic data sets, discussion about the convergenc

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

Orthogonal Joint Sparse NMF for Microarray Data Analysis

The 3D microarrays, generally known as gene-sample-time microarrays, couple the information on different time points collected by 2D microarrays that measure gene expression levels among different samples. Their analysis is useful in several biomedical applications, like monitoring dose or drug treatment responses of patients over time in pharmacogenomics studies. Many statistical and data analysis tools have been used to extract useful information. In particular, nonnegative matrix factorization (NMF), with its natural nonnegativity constraints, has demonstrated its ability to extract from 2D microarrays relevant information on specific genes involved in the particular biological process. In this paper, we propose a new NMF model, namely Orthogonal Joint Sparse NMF, to extract relevant information from 3D microarrays containing the time evolution of a 2D microarray, by adding additional constraints to enforce important biological proprieties useful for further biological analysis. We develop multiplicative updates rules that decrease the objective function monotonically, and compare our approach to state-of-the-art NMF algorithms on both synthetic and real data sets

Using underapproximations for sparse nonnegative matrix factorization

Nonnegative Matrix Factorization (NMF) has gathered a lot of attention in the last decade and has been successfully applied in numerous applications. It consists in the factorization of a nonnegative matrix by the product of two low-rank nonnegative matrices:. MªVW. In this paper, we attempt to solve NMF problems in a recursive way. In order to do that, we introduce a new variant called Nonnegative Matrix Underapproximation (NMU) by adding the upper bound constraint VW£M. Besides enabling a recursive procedure for NMF, these inequalities make NMU particularly well suited to achieve a sparse representation, improving the part-based decomposition. Although NMU is NP-hard (which we prove using its equivalence with the maximum edge biclique problem in bipartite graphs), we present two approaches to solve it: a method based on convex reformulations and a method based on Lagrangian relaxation. Finally, we provide some encouraging numerical results for image processing applications.nonnegative matrix factorization, underapproximation, maximum edge biclique problem, sparsity, image processing