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

Probability measures on infinite-dimensional Stiefel manifolds

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.

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.

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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