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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

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

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

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

[12]

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

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[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.

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 $d_{{\text{Sym}} ^+ M}\left(\sigma, q^{-1}(x_i)\right)$.
[1]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072

[2]

Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225

[3]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure and Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[4]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155

[5]

Lucas Dahinden, Álvaro del Pino. Introducing sub-Riemannian and sub-Finsler billiards. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3187-3232. doi: 10.3934/dcds.2022014

[6]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[7]

Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control and Related Fields, 2020, 10 (3) : 493-526. doi: 10.3934/mcrf.2020008

[8]

Seunghee Lee, Ganguk Hwang. A new analytical model for optimized cognitive radio networks based on stochastic geometry. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1883-1899. doi: 10.3934/jimo.2017023

[9]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations and Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193

[10]

Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 161-177. doi: 10.3934/dcdss.2021014

[11]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[12]

Beatrice Abbondanza, Stefano Biagi. Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3161-3192. doi: 10.3934/cpaa.2021101

[13]

Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control and Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041

[14]

Juan C. Cortés, Sandra E. Delgadillo-Alemán, Roberto A. Kú-Carrillo, Rafael J. Villanueva. Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022079

[15]

Raphael Kruse, Yue Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3475-3502. doi: 10.3934/dcdsb.2018253

[16]

Miguel-C. Muñoz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934/jgm.2018017

[17]

Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic and Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049

[18]

Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003

[19]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic and Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713

[20]

Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure and Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (187)
  • HTML views (59)
  • Cited by (6)

Other articles
by authors

[Back to Top]