  Principal component analysis (PCA) is very popular to perform dimension
reduction. The selection of the number of significant components is essential
but often based on some practical heuristics depending on the application. Only
few works have proposed a probabilistic approach able to infer the number of
significant components. To this purpose, this paper introduces a Bayesian
nonparametric principal component analysis (BNP-PCA). The proposed model
projects observations onto a random orthogonal basis which is assigned a prior
distribution defined on the Stiefel manifold. The prior on factor scores
involves an Indian buffet process to model the uncertainty related to the
number of components. The parameters of interest as well as the nuisance
parameters are finally inferred within a fully Bayesian framework via Monte
Carlo sampling. A study of the (in-)consistence of the marginal maximum a
posteriori estimator of the latent dimension is carried out. A new estimator of
the subspace dimension is proposed. Moreover, for sake of statistical
significance, a Kolmogorov-Smirnov test based on the posterior distribution of
the principal components is used to refine this estimate.
  The behaviour of the algorithm is first studied on various synthetic
examples. Finally, the proposed BNP dimension reduction approach is shown to be
easily yet efficiently coupled with clustering or latent factor models within a
unique framework.
