Intrinsic Dimension Estimation by Maximum Likelihood in Isotropic Probabilistic PCA

Charles Bouveyron 1 Gilles Celeux 2 Stephane Girard 3
1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 SELECT - Model selection in statistical learning
Inria Saclay - Ile de France, LMO - Laboratoire de Mathématiques d'Orsay, CNRS - Centre National de la Recherche Scientifique : UMR
3 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : A central issue in dimension reduction is choosing a sensible number of dimensions to be retained. This work demonstrates the surprising result of the asymptotic consistency of the maximum likelihood criterion for determining the intrinsic dimension of a dataset in an isotropic version of Probabilistic Principal Component Analysis (PPCA). Numerical experiments on simulated and real datasets show that the maximum likelihood criterion can actually be used in practice and outperforms existing intrinsic dimension selection criteria in various situations. This paper exhibits and outlines the limits of the maximum likelihood criterion. It leads to recommend the use of the AIC criterion in specific situations. A useful application of this work would be the automatic selection of intrinsic dimensions in mixtures of isotropic PPCA for classification.
Keywords : asymptotic consistency Probabilistic PCA isotropic model dimension reduction intrinsic dimension maximum likelihood asymptotic consistency.
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Charles Bouveyron, Gilles Celeux, Stephane Girard. Intrinsic Dimension Estimation by Maximum Likelihood in Isotropic Probabilistic PCA. Pattern Recognition Letters, Elsevier, 2011, 32 (14), pp.1706-1713. ⟨10.1016/j.patrec.2011.07.017⟩. ⟨hal-00440372v3⟩

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