Geometric Invariance of Covariance Matrices for Unsupervised Clustering, Segmentation and Action Discovery in Robotic Applications. Figueroa, N. & Billard, A. In preparation for re-submission to the Journal of Machine Learning Research (JMLR), 2019.
abstract   bibtex   
In this paper, we introduce a novel distance in the space of covariance matrices that is invariant to geometric non-deforming transformations. We refer to it as the Spectral Polytope Covariance Matrix (SPCM) distance. We prove that it is a semi-metric capable of measuring the similarity between shapes of Gaussian distributions, covariance matrices and ellipsoids; all of which are symmetric positive definite (SPD) matrices. Such a distance is desirable in data-driven robotics applications where data represented by SPD matrices must be grouped or clustered, yet is collected in different unknown frames of reference or with different timing and scale. We thus propose a geometric invariant clustering approach which leverages the SPCM distance with vector space embeddings for SPD matrices and distance-dependent Bayesian non-parametric mixture model. We show that our proposed geometric invariant clustering algorithm outperforms state-of-the-art SPD matrix clustering schemes on a variety of robotics datasets. Further, we offer an algorithmic coupling between our clustering scheme and a Bayesian non-parametric Hidden Markov Model. This coupling allows for automatic segmentation and similar action discovery in sequential tasks demonstrated from unconstrained human motions encompassing wiping, polishing, dough rolling and peeling.
@MISC{Figueroa:JMLR:2019,
  author       = {Figueroa, N. and Billard, A.},
  title        = {Geometric Invariance of Covariance Matrices for Unsupervised Clustering, Segmentation and Action Discovery in Robotic Applications},
  howpublished = {In preparation for re-submission to the Journal of Machine Learning Research (JMLR)},
  year         = {2019},
  abstract = {In this paper, we introduce a novel distance in the space of covariance matrices that is invariant to geometric non-deforming transformations. We refer to it as the Spectral Polytope Covariance Matrix (SPCM) distance. We prove that it is a semi-metric capable of measuring the similarity between shapes of Gaussian distributions, covariance matrices and ellipsoids; all of which are symmetric positive definite (SPD) matrices. Such a distance is desirable in data-driven robotics applications where data represented by SPD matrices must be grouped or clustered, yet is collected in different unknown frames of reference or with different timing and scale. We thus propose a geometric invariant clustering approach which leverages the SPCM distance with vector space embeddings for SPD matrices and distance-dependent Bayesian non-parametric mixture model. We show that our proposed geometric invariant clustering algorithm outperforms state-of-the-art SPD matrix clustering schemes on a variety of robotics datasets. Further, we offer an algorithmic coupling between our clustering scheme and a Bayesian non-parametric Hidden Markov Model. This coupling allows for automatic segmentation and similar action discovery in sequential tasks demonstrated from unconstrained human motions encompassing wiping, polishing, dough rolling and peeling.}
}
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