2019/01/30 by Nicolas Gillis, Le Thi Khanh Hien, Gillis, Nicolas +5 · 1 citation
Computer Science · Decision Sciences · #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimal Experimental Design Methods #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1901.10757
openalex publication_date 2019/01/30 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28
Nonnegative matrix factorization (NMF) is a linear dimensionality reduction\ntechnique for analyzing nonnegative data. A key aspect of NMF is the choice of\nthe objective function that depends on the noise model (or statistics of the\nnoise) assumed on the data. In many applications, the noise model is unknown\nand difficult to estimate. In this paper, we define a multi-objective NMF\n(MO-NMF) problem, where several objectives are combined within the same NMF\nmodel. We propose to use Lagrange duality to judiciously optimize for a set of\nweights to be used within the framework of the weighted-sum approach, that is,\nwe minimize a single objective function which is a weighted sum of the all\nobjective functions. We design a simple algorithm based on multiplicative\nupdates to minimize this weighted sum. We show how this can be used to find\ndistributionally robust NMF (DR-NMF) solutions, that is, solutions that\nminimize the largest error among all objectives, using a dual approach solved\nvia a heuristic inspired from the Frank-Wolfe algorithm. We illustrate the\neffectiveness of this approach on synthetic, document and audio data sets. The\nresults show that DR-NMF is robust to our incognizance of the noise model of\nthe NMF problem.\n