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September  2017, 9(3): 291-316. doi: 10.3934/jgm.2017012

Probability measures on infinite-dimensional Stiefel manifolds

Eleonora Bardelli 1,‡, and  Andrea Carlo Giuseppe Mennucci 2,,

1. 

Microsoft Deutschland GmbH, Germany
2. 

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italia

* Corresponding author: A. C. G. Mennucci

‡ Eleonora Bardelli contributed to this paper in her personal capacity. The views expressed in this paper are those of the authors and do not necessarily reflect the views of Microsoft Corporation

Received  December 2015 Revised  June 2016 Published  June 2017

An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds.

Suppose that $H$ is an infinite-dimensional separable Hilbert space.

Let $S\subset H$ be the sphere, $p\in S$. Let $\mu$ be the push forward of a Gaussian measure $\gamma$ from $T_p S$ onto $S$ using the exponential map. Let $v\in T_p S$ be a Cameron-Martin vector for $\gamma$; let $R$ be a rotation of $S$ in the direction $v$, and $\nu=R_\# \mu$ be the rotated measure. Then $\mu,\nu$ are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron-Martin direction produces equivalent measures.

Let $\gamma$ be a Gaussian measure on $H$; then there exists a smooth closed manifold $M\subset H$ such that the projection of $H$ to the nearest point on $M$ is not well defined for points in a set of positive $\gamma$ measure.

Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability.

Keywords: Probability, Gaussian measure, infinite-dimensional manifold, Riemannian manifold, Hilbert space, Shape Space.
Mathematics Subject Classification: Primary: 60D05, 58D15, 58B20; Secondary: 46C05, 54C56.
Citation: Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

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D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

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G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer, 2006. doi: 10.1007/3-540-29021-4.  Google Scholar

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M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

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A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. , 20 (1999), 303-353, URL http://dx.doi.org/10.1137/S0895479895290954. doi: 10.1137/S0895479895290954.  Google Scholar

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H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

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N. Grossman, Hilbert manifolds without epiconjugate points, Proc. Amer. Math. Soc., 16 (1965), 1365-1371.  doi: 10.1090/S0002-9939-1965-0188943-7.  Google Scholar

[12]

P. Harms and A. Mennucci, Geodesics in infinite dimensional Stiefel and Grassmann manifolds, Comptes rendus -Mathématique, 350 (2012), 773-776, URL http://cvgmt.sns.it/paper/336/. doi: 10.1016/j.crma.2012.08.010.  Google Scholar

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D. G. Kendall, The diffusion of shape, Advances in applied probability, (), 428-430.   Google Scholar

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D. G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces, Bulletin of the London Mathematical Society, 16 (1984), 81-121.  doi: 10.1112/blms/16.2.81.  Google Scholar

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A. N. Kolmogorov, La transformation de laplace dans les espaces linéaires, CR Acad. Sci. Paris, 200 (1935), 1717-1718.   Google Scholar

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S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar

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H. Le and D. G. Kendall, The Riemannian structure of Euclidean shape spaces: A novel environment for statistics, The Annals of Statistics, 21 (1993), 1225-1271.  doi: 10.1214/aos/1176349259.  Google Scholar

[18]

C. Mantegazza and A. C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Applied Math. and Optim., 47 (2003), 1-25.  doi: 10.1007/s00245-002-0736-4.  Google Scholar

[19]

A. C. G. Mennucci, Regularity and variationality of solutions to Hamilton-Jacobi equations. Ⅰ. Regularity, ESAIM Control Optim. Calc. Var. , 10 (2004), 426-451 (electronic), URL http://dx.doi.org/10.1051/cocv:2004014. doi: 10.1051/cocv:2004014.  Google Scholar

[20]

G. Sundaramoorthi, A. Mennucci, S. Soatto and A. Yezzi, A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering, SIAM J. Imaging Sci. , 4 (2011), 109-145, URL http://dx.doi.org/10.1137/090781139. doi: 10.1137/090781139.  Google Scholar

[21]

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math. , 58 (1998), 565-586 (electronic), URL http://dx.doi.org/10.1137/S0036139995287685. doi: 10.1137/S0036139995287685.  Google Scholar

[22]

L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. , 19 (2008), 25-57, URL http://dx.doi.org/10.4171/RLM/506. doi: 10.4171/RLM/506.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

C. J. Atkin, The Hopf-Rinow theorem is false in infinite dimensions, Bull. London Math. Soc., 7 (1975), 261-266.  doi: 10.1112/blms/7.3.261.  Google Scholar

[3]

V. I. Bogachev, Measure Theory. Vol. Ⅰ, Ⅱ, Springer-Verlag, Berlin, 2007, URL http://dx.doi.org/10.1007/978-3-540-34514-5. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[4]

V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[5]

L. Breiman, Probability, Addison-Wesley, 1968.  Google Scholar

[6]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

[7]

G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer, 2006. doi: 10.1007/3-540-29021-4.  Google Scholar

[8]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

[9]

A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. , 20 (1999), 303-353, URL http://dx.doi.org/10.1137/S0895479895290954. doi: 10.1137/S0895479895290954.  Google Scholar

[10]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[11]

N. Grossman, Hilbert manifolds without epiconjugate points, Proc. Amer. Math. Soc., 16 (1965), 1365-1371.  doi: 10.1090/S0002-9939-1965-0188943-7.  Google Scholar

[12]

P. Harms and A. Mennucci, Geodesics in infinite dimensional Stiefel and Grassmann manifolds, Comptes rendus -Mathématique, 350 (2012), 773-776, URL http://cvgmt.sns.it/paper/336/. doi: 10.1016/j.crma.2012.08.010.  Google Scholar

[13]

D. G. Kendall, The diffusion of shape, Advances in applied probability, (), 428-430.   Google Scholar

[14]

D. G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces, Bulletin of the London Mathematical Society, 16 (1984), 81-121.  doi: 10.1112/blms/16.2.81.  Google Scholar

[15]

A. N. Kolmogorov, La transformation de laplace dans les espaces linéaires, CR Acad. Sci. Paris, 200 (1935), 1717-1718.   Google Scholar

[16]

S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[17]

H. Le and D. G. Kendall, The Riemannian structure of Euclidean shape spaces: A novel environment for statistics, The Annals of Statistics, 21 (1993), 1225-1271.  doi: 10.1214/aos/1176349259.  Google Scholar

[18]

C. Mantegazza and A. C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Applied Math. and Optim., 47 (2003), 1-25.  doi: 10.1007/s00245-002-0736-4.  Google Scholar

[19]

A. C. G. Mennucci, Regularity and variationality of solutions to Hamilton-Jacobi equations. Ⅰ. Regularity, ESAIM Control Optim. Calc. Var. , 10 (2004), 426-451 (electronic), URL http://dx.doi.org/10.1051/cocv:2004014. doi: 10.1051/cocv:2004014.  Google Scholar

[20]

G. Sundaramoorthi, A. Mennucci, S. Soatto and A. Yezzi, A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering, SIAM J. Imaging Sci. , 4 (2011), 109-145, URL http://dx.doi.org/10.1137/090781139. doi: 10.1137/090781139.  Google Scholar

[21]

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math. , 58 (1998), 565-586 (electronic), URL http://dx.doi.org/10.1137/S0036139995287685. doi: 10.1137/S0036139995287685.  Google Scholar

[22]

L. Younes, P. W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. , 19 (2008), 25-57, URL http://dx.doi.org/10.4171/RLM/506. doi: 10.4171/RLM/506.  Google Scholar

Figure 1.  Proof of Proposition 3.6. Two points on the ellipsoid $C$, their images under $\exp_{-p}$ and, in white, their images under $\exp_p$. The black diamonds on the sphere can coincide with the white diamonds only if they all lie on the equator.
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