Abstract
We present an algorithm for grouping families of probability density functions (pdfs). We exploit the fact that under the square-root re-parametrization, the space of pdfs forms a Riemannian manifold, namely the unit Hilbert sphere. An immediate consequence of this re-parametrization is that different families of pdfs form different submanifolds of the unit Hilbert sphere. Therefore, the problem of clustering pdfs reduces to the problem of clustering multiple submanifolds on the unit Hilbert sphere. We solve this problem by first learning a low-dimensional representation of the pdfs using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces. Then, by assuming that the pdfs from different groups are separated, we show that the null space of a matrix built from the local representation gives the segmentation of the pdfs. We also apply of our approach to the texture segmentation problem in computer vision.
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Rao, C.R.: Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945)
Amari, S.: Differential-Geometrical Methods in Statistics. Springer, Heidelberg (1985)
Mio, W., Badlyans, D., Liu, X.: A computational approach to Fisher information geometry with applications to image analysis. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 18–33. Springer, Heidelberg (2005)
Srivastava, A., Joshi, S., Mio, W., Liu, X.: Statistical shape analysis: clustering, learning, and testing. IEEE Transactions on PAMI 27(4), 590–602 (2005)
Srivastava, A., Jermyn, I., Joshi, S.: Riemannian analysis of probability density functions with applications in vision. In: IEEE CVPR (2007)
Hotelling, H.: Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 24, 417–441 (1933)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge (2002)
Roweis, S., Saul, L.: Think globally, fit locally: Unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research 4, 119–155 (2003)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Neural Information Processing Systems, pp. 585–591 (2002)
Donoho, D., Grimes, C.: Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. PNAS 100(10), 5591–5596 (2003)
Goh, A., Vidal, R.: Clustering and dimensionality reduction on Riemannian manifolds. In: IEEE CVPR (2008)
do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)
Frechet, M.: Les elements aleatoires de nature quelconque dans un espace distancie. Annales De L’Institut Henri Poincare 10, 235–310 (1948)
Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87(2) (2007)
Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. International Journal of Computer Vision 66(1), 41–46 (2006)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30(5), 509–541 (1977)
Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Cencov, N.N.: Statistical decision rules and optimal inference. In: Translations of Mathematical Monographs, vol. 53. AMS (1982)
Varma, M., Zisserman, A.: A statistical approach to texture classification from single images. International Journal of Computer Vision 62(1-2), 61–81 (2005)
Schmid, C.: Constructing models for content-based image retrieval. In: IEEE Conference on Computer Vision and Pattern Recognition (2001)
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Goh, A., Vidal, R. (2008). Unsupervised Riemannian Clustering of Probability Density Functions. In: Daelemans, W., Goethals, B., Morik, K. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2008. Lecture Notes in Computer Science(), vol 5211. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87479-9_43
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DOI: https://doi.org/10.1007/978-3-540-87479-9_43
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