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

Authors:
Kun Huang;Yi Ma;René Vidal
Affiliations:
Dept. of Electrical & Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL;Dept. of Electrical & Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL;Dept. of Biomedical Engineering, Johns Hopkins University, Baltimore, MD
Venue:
CVPR'04 Proceedings of the 2004 IEEE computer society conference on Computer vision and pattern recognition
Year:
2004

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Abstract

In this paper, we propose a robust model selection criterion for mixtures of subspaces 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 subspaces, 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 noniterative 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.