Published Paper

Inserted: 23 nov 2015
Last Updated: 21 jun 2017

Journal: Journal of Geometric Mechanics
Volume: 9
Number: 3
Pages: 291-316
Year: 2017
Doi: 10.3934/jgm.2017012
Notes:

special issue on "Infinite-Dimensional Riemannian Geometry"

Abstract:

An interest in infinite dimensional manifolds has recently appeared in Shape Theory. One such 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; this has many applications, such as object recognition, estimation, tracking, etc. In the case of finite dimensional manifolds, there is a vast literature regarding the definition and perusal of probabilities on finite dimensional manifolds. Unfortunately less is know about the infinite dimensional case. In this paper we will present some negative and some positive results. We highlight the main results in this abstract. Suppose in the following that $H$ is an infinite dimensional separable Hilbert space.

Let $S\subset H$ be the sphere, fix $p\in S$. Let $\mu$ be the probability that results when wrapping a Gaussian measure $\gamma$ from $T_pS$ onto $S$ using the exponential map. Let $v\in T_pS$ 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 when $\gamma$ is a Gaussian measure on $H$ and $T$ is the translation in a Cameron--Martin direction, then $T_\#\gamma$ and $\gamma$ are mutually absolutely continuous.

Suppose now that $\gamma$ is 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. This is opposite to what is observed in finite dimensional spaces.

The situation is instead better for a special class of smooth manifolds, the Stiefel manifolds. Let $M=\mathrm{St}({n},{H})\subset H^n$ be the Stiefel manifold. Let $\gamma$ be a non-degenerate Gaussian measure in $H^n$; then the projection of $x\in H^n$ to the nearest point $z\in M$ is well defined for $\gamma$-almost all $x$. Consequently it is possible to project $\gamma$ to $M$ to define a probability on $M$. This has important applications for Shape Theory, since $\mathrm{St}({2},{L^2})$ has been proposed as the model for the manifold of all immersed curves in the plane. The above procedure can be easily implemented numerically, and provides an effective family of probability models on the space of immersed curves in the plane.

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