
-
Previous Article
Complete spelling rules for the Monster tower over three-space
- JGM Home
- This Issue
-
Next Article
About simple variational splines from the Hamiltonian viewpoint
Probability measures on infinite-dimensional Stiefel manifolds
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.
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. 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. |
[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. |
[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. 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. |
[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. |
[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. |

[1] |
Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020463 |
[2] |
Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230 |
[3] |
Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048 |
[4] |
Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 |
[5] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
[6] |
Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 |
[7] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
[8] |
Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020031 |
[9] |
Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 |
[10] |
Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169 |
[11] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[12] |
Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 |
[13] |
Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020459 |
[14] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[15] |
Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021003 |
[16] |
Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021012 |
[17] |
Bing Liu, Ming Zhou. Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin. Journal of Industrial & Management Optimization, 2021, 17 (2) : 937-952. doi: 10.3934/jimo.2020005 |
[18] |
Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 |
[19] |
Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071 |
[20] |
Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]