Minimum effective dimension for mixtures of subspaces: A robust GPCA algorithm and its applications

Abstract

In this paper, we propose a robust model selection criterion for mixtures ofsubspaces called minimum effective dimension (MED). Previous information-theoretic model selection criteria typically assume that data can be modelled with a parametric model of certain (possibly differing) dimension and a known error distribution. However, for mixtures of subspaces with different dimensions, a generalized notion of dimensionality is needed and hence introduced in this paper. The proposed MED criterion minimizes this geometric dimension subject to a given error tolerance (regardless of the noise distribution). Furthermore, combined with a purely algebraic approach to clustering mixtures of sub-spaces, namely the Generalized PCA (GPCA), the MED is designed to also respect the global algebraic and geometric structure of the data. The result is a non-iterative algorithm called robust GPCA that estimates from noisy data an unknown number of subspaces with unknown and possibly different dimensions subject to a maximum error bound. We test the algorithm on synthetic noisy data and in applications such as motion/image/video segmentation.