Seminar Details
2020-11-10 (14h) : Algorithms for Nonnegative Matrix Factorization
At MS Teams
Organized by Mathematical Engineering
Speaker :
Leplat, Valentin
Abstract :
The low-rank approximation of a matrix is a key problem in data analysis, and is equivalent to linear dimensionality reduction (LDR). LDR techniques such as principal component analysis are powerful tools for the analysis of high-dimensional data. In this talk, we consider a particular low-rank approximation model, namely nonnegative matrix factorization (NMF). NMF is the problem of approximating a given nonnegative matrix, V, with the product of two nonnegative matrices, W and H. The nonnegativity constraint on W and H allows us to extract easily interpretable and meaningful information from the input data. However, they make the problem much more difficult to solve (NP-hard). This talk focuses on the algorithmic aspect to compute an NMF for an input matrix. First, we introduce NMF models and algorithms. Second, we introduce a general framework to design algorithms (namely multiplicative updates) for NMF problems minimizing the β-divergence between V and WH, with disjoint equality constraints and penalty terms in the objective function. Finally, we introduce a novel framework using tools from conic optimization in order to compute an NMF which is exact, that is, with V = WH.
The content of this talk contains joint works with Nicolas Gillis (UMons), Jérôme Idier (Ecole Centrale de Nantes), Yurii Nesterov (UCLouvain) and François Glineur (UCLouvain).
Keywords : nonnegative matrix factorization (NMF), β-divergences, disjoint constraints, exact NMF, conic programming, successive approximations.