publication . Preprint . 2017

Transform-Invariant Non-Parametric Clustering of Covariance Matrices and its Application to Unsupervised Joint Segmentation and Action Discovery

Figueroa, Nadia; Billard, Aude;
Open Access English
  • Published: 27 Oct 2017
Abstract
In this work, we tackle the problem of transform-invariant unsupervised learning in the space of Covariance matrices and applications thereof. We begin by introducing the Spectral Polytope Covariance Matrix (SPCM) Similarity function; a similarity function for Covariance matrices, invariant to any type of transformation. We then derive the SPCM-CRP mixture model, a transform-invariant non-parametric clustering approach for Covariance matrices that leverages the proposed similarity function, spectral embedding and the distance-dependent Chinese Restaurant Process (dd-CRP) (Blei and Frazier, 2011). The scalability and applicability of these two contributions is ex...
Subjects
free text keywords: Computer Science - Learning
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R. Vemulapalli, J. K. Pillai, and R. Chellappa. Kernel learning for extrinsic classi cation of manifold features. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pages 1782{1789, June 2013.

Ulrike von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4): 395{416, 2007.

L. Wang, J. Zhang, L. Zhou, C. Tang, and W. Li. Beyond covariance: Feature representation with nonlinear kernel matrices. In 2015 IEEE International Conference on Computer Vision (ICCV), pages 4570{4578, Dec 2015.

Lihi Zelnik-manor and Pietro Perona. Self-tuning spectral clustering. In Advances in Neural Information Processing Systems 17, pages 1601{1608. MIT Press, 2004.

Abstract
In this work, we tackle the problem of transform-invariant unsupervised learning in the space of Covariance matrices and applications thereof. We begin by introducing the Spectral Polytope Covariance Matrix (SPCM) Similarity function; a similarity function for Covariance matrices, invariant to any type of transformation. We then derive the SPCM-CRP mixture model, a transform-invariant non-parametric clustering approach for Covariance matrices that leverages the proposed similarity function, spectral embedding and the distance-dependent Chinese Restaurant Process (dd-CRP) (Blei and Frazier, 2011). The scalability and applicability of these two contributions is ex...
Subjects
free text keywords: Computer Science - Learning
Download from

Brenna Argall, Sonia Chernova, Manuela M. Veloso, and Brett Browning. A survey of robot learning from demonstration. Robotics and Autonomous Systems, 57(5):469{483, 2009.

V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Log-Euclidean metrics for fast and simple calculus on di usion tensors. Magnetic Resonance in Medicine, 56(2):411{421, 2006a. [OpenAIRE]

Samuel J. Gershman and David M. Blei. A tutorial on bayesian nonparametric models. Journal of Mathematical Psychology, 56(1):1 { 12, 2012.

Soumya Ghosh, Erik Sudderth, Matthew Loper, and Michael Black. From deformations to parts: Motion-based segmentation of 3D objects. In Advances in Neural Information Processing Systems 25 (NIPS), pages 2006{2014. MIT Press, 2012.

V. Kruger, V. Tikhano , L. Natale, and G. Sandini. Imitation learning of non-linear pointto-point robot motions using dirichlet processes. In 2012 IEEE International Conference on Robotics and Automation, pages 2029{2034, May 2012.

A. L. Pais, K. Umezawa, Y. Nakamura, and A. Billard. Task parametrization using continuous constraints extracted from human demonstrations. Accepted, IEEE TRO, 2015. [OpenAIRE]

X. Pennec, P. Fillard, and N. Ayache. A riemannian framework for tensor computing. Int. J. Comput. Vision, 66(1):41{66, January 2006. ISSN 0920-5691.

Oncel Tuzel, Fatih Porikli, and Peter Meer. Region covariance: A fast descriptor for detection and classi cation. In Proceedings of the 9th European Conference on Computer Vision - Volume Part II, ECCV'06, pages 589{600, Berlin, Heidelberg, 2006. SpringerVerlag.

R. Vemulapalli, J. K. Pillai, and R. Chellappa. Kernel learning for extrinsic classi cation of manifold features. In Computer Vision and Pattern Recognition (CVPR), 2013 IEEE Conference on, pages 1782{1789, June 2013.

Ulrike von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4): 395{416, 2007.

L. Wang, J. Zhang, L. Zhou, C. Tang, and W. Li. Beyond covariance: Feature representation with nonlinear kernel matrices. In 2015 IEEE International Conference on Computer Vision (ICCV), pages 4570{4578, Dec 2015.

Lihi Zelnik-manor and Pietro Perona. Self-tuning spectral clustering. In Advances in Neural Information Processing Systems 17, pages 1601{1608. MIT Press, 2004.

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