Local PCA learning with resolution-dependent mixtures of Gaussians

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Abstract
A globally linear model, as implied by conven- tional Principal Component Analysis (PCA), may be insufficient to represent multivariate data in many situations. It has been known for some time that a combination of several "lo- cal" PCA's can provide a suitable approach in such cases (1, 2). An important question is then how to find an appropriate partitioning of the data space together with a proper choice of the local numbers of principal components (PC's). In this contribution we address both problems within a density estimation framework and pro- pose a probabilistic approach which is based on a mixture of subspace-constrained Gaussians. Thereby the number of local PC's depends on a global resolution parameter, which represents the assumed noise level and determines the de- gree of smoothing imposed by the model. As a consequence the model leads to an automatic resolution-dependent adjustment of the optimal principal subspace dimensionalities, which may vary among the different mixture components. Furthermore it allows to provide the optimiza- tion with an annealing scheme, which solves the initialization problem and offers an incremen- tal model refinement procedure. Experimental results on synthetic and high-dimensional real- world data illustrate the merits of the proposed approach.

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