Page 12 - Contributed Paper Session (CPS) - Volume 4

CPS2101 Bertail Patrice et al.

Nonnegative matrix factorization: A semi-
parametric statistical view and selection model
2
1
Bertail Patrice , Cl´emen¸con St´ephane , Zetlaoui M´elanie 1
1 Paris Nanterre University, Nanterre, France,
2 T´el´ecom ParisTech, Paris, France

Abstract
The goal of Nonnegative Matrix Factorization (NMF) consists in finding a
convex cone in the positive orthant,” representing accurately” a cloud of
multivariate nonnegative data. The dimension of the convex cone is assumed
to be smaller than the dimension of the data space. Whereas the majority of
the literature dedicated to NMF focused on algorithmic issues related to the
computation of representations maximizing some goodness-of-fit criterion,
statistical grounds for such M-estimation techniques have not been exhibited
yet. Here, we investigate the semiparametric framework: through the
specification of a variety of probabilistic generative models and under
statistical identifiability assumptions and we can construct a Z-estimator with
estimated nuisance parameters based on the efficient score. Under
appropriate assumptions, this Z-estimator yields asymptotically normal
estimates of C’s rays. In this context, model selection issues related to the
dimension of the underlying cone C are considered through the AIC and BIC
approaches. We show, under regularity assumptions, that we can recover the
optimal number of C’s rays.

Keywords
Nonnegative matrix factorization; latent variable model; semiparametric
estimation; identifiability; model selection; efficient scores.

1. Introduction
In a wide variety of applications, data are nonnegative by nature: pixel
intensities, amplitude spectra, occurrence counts, food consumption, user
scores, stock market values, etc. Nonnegative matrix factorization (NMF)
precisely aims at finding (linearly independent) latent vectors with
nonnegative coordinates, of which observations can be viewed as convex
linear combinations. Originally proposed by [7] in the context of facial images
analysis, NMF has recently received a good deal of attention in the fields of
machine learning and signal/image processing and has been applied to a
variety of applications in different fields.
Whereas the design of NMF computational techniques has been the
subject of intense research these last few years in the signal processing and

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