| 1. | Microsoft Deutschland GmbH, Germany |
| 2. | Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italia |
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.
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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. |
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C. Mantegazza and A. C. Mennucci,
Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Applied Math. and Optim., 47 (2003), 1-25.
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G. Sundaramoorthi, A. Mennucci, S. Soatto and A. Yezzi,
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SIAM J. Imaging Sci. , 4 (2011), 109-145, URL http://dx.doi.org/10.1137/090781139.
doi: 10.1137/090781139. |
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doi: 10.1137/S0036139995287685. |
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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
| [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. |
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