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CPS2101 Bertail Patrice et al.

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

                  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.

                  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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