Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

Sommer Stefan; Svane Anne Marie

doi:10.3934/jgm.2017015

Article Contents

2017, Volume 9, Issue 3: 391-410. Doi: 10.3934/jgm.2017015

This issue Previous Article Geometry of matrix decompositions seen through optimal transport and information geometry Next Article

Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

Sommer Stefan^1, and
Svane Anne Marie^2,

1.
University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen E, Denmark
2.
Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

Received: December 31, 2015

Revised: May 31, 2016

Published: September 2017

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Abstract

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hörmander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

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Mathematics Subject Classification: Primary: 53C17, 60E05; Secondary: 62H11.

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Figure 1. Sampled data on an ellipsoid realized as endpoints of sample paths of the process $X_t$ (black lines and points). Mean $x_0=\pi(u_0)$ (green point) and covariance $\sigma^{-1}$ (ellipsis over mean) are estimated by minimizing (15). The most probable paths for the driving process connects $x_0$ and the sample paths (gray lines) and minimize the distances $d_{{\text{Sym}} ^+ M}\left(\sigma, q^{-1}(x_i)\right)$.

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References

	A. A. Agrachev , Any sub-Riemannian metric has points of smoothness, Doklady Mathematics, 79 (2009) , 45-47. doi: 10.1134/S106456240901013X.
	D. Barilari , U. Boscain and R. W. Neel , Small-time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012) , 373-416. doi: 10.4310/jdg/1354110195.
	R. L. Bryant, A survey of Riemannian metrics with special holonomy groups, in Proceedings of the International Congress of Mathematicians, vol. 1, 2, Amer. Math. Soc., Berkeley, California, 1987,505-514.
	M. Emery, Stochastic Calculus in Manifolds, Universitext, Springer Berlin Heidelberg, Berlin, Heidelberg, 1989. doi: 10.1007/978-3-642-75051-9.
	P. Fletcher , C. Lu and S. Joshi , Statistics of shape via principal geodesic analysis on Lie groups, CVPR, 1 (2003) , I95-I101. doi: 10.1109/CVPR.2003.1211342.
	M. Fréchet , Les éléments aléatoires de nature quelconque dans un espace distancie, Ann. Inst. H. Poincaré, 10 (1948) , 215-310.
	T. Fujita and S.-i. Kotani , The Onsager-Machlup function for diffusion processes, Journal of Mathematics of Kyoto University, 22 (1982) , 115-130. doi: 10.1215/kjm/1250521863.
	U. Grenander and M. I. Miller , Computational anatomy: An emerging discipline, Q. Appl. Math., 56 (1998) , 617-694. doi: 10.1090/qam/1668732.
	E. P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Soc., 2002. doi: 10.1090/gsm/038.
	D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.
	R. Léandre , Minoration en temps petit de la densité d'une diffusion dégénérée, Journal of Functional Analysis, 74 (1987) , 399-414. doi: 10.1016/0022-1236(87)90031-0.
	P. W. Michor, Topics in Differential Geometry, American Mathematical Soc., 2008. doi: 10.1090/gsm/093.
	K. Modin and S. Sommer, Proceedings of Math On The Rocks Shape Analysis Workshop in Grundsund 2015, URL http://dx.doi.org/10.5281/zenodo.33558.
	K.-P. Mok , On the differential geometry of frame bundles of Riemannian manifolds, Journal Fur Die Reine Und Angewandte Mathematik, 302 (1978) , 16-31. doi: 10.1515/crll.1978.302.16.
	R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Soc., Providence, RI, 2002.
	X. Pennec , Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements, J. Math. Imaging Vis., 25 (2006) , 127-154. doi: 10.1007/s10851-006-6228-4.
	S. Sommer, Diffusion Processes and PCA on Manifolds, Mathematisches Forschungsinstitut Oberwolfach, 2014, URL https://www.mfo.de/document/1440a/OWR_2014_44.pdf.
	S. Sommer , Anisotropic distributions on manifolds: Template estimation and most probable paths, Information Processing in Medical Imaging, (2015) , 193-204. doi: 10.1007/978-3-319-19992-4_15.
	S. Sommer, Evolution equations with anisotropic distributions and diffusion PCA, in Geometric Science of Information (eds. F. Nielsen and F. Barbaresco), Lecture Notes in Computer Science, Springer International Publishing, 9389 (2015), 3-11. doi: 10.1007/978-3-319-25040-3_1.
	R. S. Strichartz , Sub-Riemannian geometry, Journal of Differential Geometry, 24 (1986) , 221-263. doi: 10.4310/jdg/1214440436.
	H. J. Sussmann , Orbits of families of vector fields and integrability of distributions, Transactions of the American Mathematical Society, 180 (1973) , 171-188. doi: 10.1090/S0002-9947-1973-0321133-2.
	M. E. Tipping and C. M. Bishop , Probabilistic principal component analysis, Journal of the Royal Statistical Society. Series B, 61 (1999) , 611-622. doi: 10.1111/1467-9868.00196.
	S. R. S. Varadhan , On the behavior of the fundamental solution of the heat equation with variable coefficients, Communications on Pure and Applied Mathematics, 20 (1967) , 431-455. doi: 10.1002/cpa.3160200210.