September  2017, 9(3): 391-410. doi: 10.3934/jgm.2017015

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

1. 

University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen E, Denmark

2. 

Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

Received  January 2016 Revised  June 2016 Published  June 2017

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.

Citation: Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015
References:
[1]

A. A. Agrachev, Any sub-Riemannian metric has points of smoothness, Doklady Mathematics, 79 (2009), 45-47.  doi: 10.1134/S106456240901013X.  Google Scholar

[2]

D. BarilariU. 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.  Google Scholar

[3]

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.  Google Scholar

[4]

M. Emery, Stochastic Calculus in Manifolds, Universitext, Springer Berlin Heidelberg, Berlin, Heidelberg, 1989. doi: 10.1007/978-3-642-75051-9.  Google Scholar

[5]

P. FletcherC. 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.  Google Scholar

[6]

M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancie, Ann. Inst. H. Poincaré, 10 (1948), 215-310.   Google Scholar

[7]

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.  Google Scholar

[8]

U. Grenander and M. I. Miller, Computational anatomy: An emerging discipline, Q. Appl. Math., 56 (1998), 617-694.  doi: 10.1090/qam/1668732.  Google Scholar

[9]

E. P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Soc., 2002. doi: 10.1090/gsm/038.  Google Scholar

[10] D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.   Google Scholar
[11]

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.  Google Scholar

[12]

P. W. Michor, Topics in Differential Geometry, American Mathematical Soc., 2008. doi: 10.1090/gsm/093.  Google Scholar

[13]

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. Google Scholar

[14]

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.  Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Soc., Providence, RI, 2002.  Google Scholar

[16]

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.  Google Scholar

[17]

S. Sommer, Diffusion Processes and PCA on Manifolds, Mathematisches Forschungsinstitut Oberwolfach, 2014, URL https://www.mfo.de/document/1440a/OWR_2014_44.pdf. Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

R. S. Strichartz, Sub-Riemannian geometry, Journal of Differential Geometry, 24 (1986), 221-263.  doi: 10.4310/jdg/1214440436.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

show all references

References:
[1]

A. A. Agrachev, Any sub-Riemannian metric has points of smoothness, Doklady Mathematics, 79 (2009), 45-47.  doi: 10.1134/S106456240901013X.  Google Scholar

[2]

D. BarilariU. 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.  Google Scholar

[3]

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.  Google Scholar

[4]

M. Emery, Stochastic Calculus in Manifolds, Universitext, Springer Berlin Heidelberg, Berlin, Heidelberg, 1989. doi: 10.1007/978-3-642-75051-9.  Google Scholar

[5]

P. FletcherC. 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.  Google Scholar

[6]

M. Fréchet, Les éléments aléatoires de nature quelconque dans un espace distancie, Ann. Inst. H. Poincaré, 10 (1948), 215-310.   Google Scholar

[7]

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.  Google Scholar

[8]

U. Grenander and M. I. Miller, Computational anatomy: An emerging discipline, Q. Appl. Math., 56 (1998), 617-694.  doi: 10.1090/qam/1668732.  Google Scholar

[9]

E. P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Soc., 2002. doi: 10.1090/gsm/038.  Google Scholar

[10] D. D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.   Google Scholar
[11]

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.  Google Scholar

[12]

P. W. Michor, Topics in Differential Geometry, American Mathematical Soc., 2008. doi: 10.1090/gsm/093.  Google Scholar

[13]

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. Google Scholar

[14]

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.  Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Soc., Providence, RI, 2002.  Google Scholar

[16]

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.  Google Scholar

[17]

S. Sommer, Diffusion Processes and PCA on Manifolds, Mathematisches Forschungsinstitut Oberwolfach, 2014, URL https://www.mfo.de/document/1440a/OWR_2014_44.pdf. Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

R. S. Strichartz, Sub-Riemannian geometry, Journal of Differential Geometry, 24 (1986), 221-263.  doi: 10.4310/jdg/1214440436.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

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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