Abstract

Data is often available in matrix form, in which columns are samples, and processing of such data often entails finding an approximate factorization of the matrix into two factors. The first factor (the “dictionary”) yields recurring patterns characteristic of the data. The second factor (“the activation matrix”) describes in which proportions each data sample is made of these patterns. Nonnegative matrix factorization (NMF) is a popular unsupervised learning technique for analysing data with nonnegative values, with applications in many areas such as in text information retrieval, recommender systems, audio signal processing, and hyperspectral imaging. In a first part, I will give a short tutorial presentation about NMF for data processing, with a focus on majorization-minimization algorithms for NMF with the beta-divergence, a continuous family of loss functions that takes the quadratic loss, KL divergence and Itakura-Saito divergence as special cases. Then, I will present applications for hyperspectral unmixing in remote sensing and factor analysis in dynamic positron emission tomography, introducing robust variants of NMF that account for outliers, nonlinear phenomena or specific binding.

References C. Févotte, J. Idier. Algorithms for nonnegative matrix factorization with the beta-divergence. Neural computation, 2011. C. Févotte, N. Dobigeon. Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization. IEEE Transactions on Image Processing, 2015. Y. C. Cavalcanti, T. Oberlin, N. Dobigeon, C. Févotte, S. Stute, M. J. Ribeiro, C. Tauber. Factor analysis of dynamic PET images: beyond Gaussian noise. IEEE Transactions on Medical Imaging, 2019. A. Marmin, H. Goulart, C. Févotte. Joint majorization-minimization for nonnegative matrix factorization with the beta-divergence. IEEE Transactions on Signal Processing, 2023.

Bio:

Cédric Févotte is a CNRS research director with Institut de Recherche en Informatique de Toulouse (IRIT). Previously, he was a CNRS researcher at Laboratoire Lagrange (Nice, 2013-2016) & Télécom ParisTech (2007-2013), a research engineer at Mist-Technologies (the startup that became Audionamix, 2006-2007) and a postdoc at University of Cambridge (2003-2006). He holds MEng and PhD degrees in EECS from École Centrale de Nantes. His research interests concern statistical signal processing and machine learning, with particular interests in matrix factorization, inverse problems, source separation and recommender systems. Selected distinctions: IEEE Fellow (2022), ERC Consolidator Grant (2016-2022), IEEE Signal Processing Society Sustained Impact Paper Award (2023), IEEE Signal Processing Society Best Paper Award (2014).