American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

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Journal of Geometric Mechanics

September 2017 , Volume 9 , Issue 3

Special issue on infinite-dimensional Riemannian geometry, Part II

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2017, 9(3): 257-290 doi: 10.3934/jgm.2017011 +[Abstract](453) +[HTML](10) +[PDF](2543.9KB)
Abstract:

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

2017, 9(3): 291-316 doi: 10.3934/jgm.2017012 +[Abstract](321) +[HTML](2) +[PDF](609.1KB)
Abstract:

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 \begin{document}$H$\end{document} is an infinite-dimensional separable Hilbert space.

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

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

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

2017, 9(3): 317-333 doi: 10.3934/jgm.2017013 +[Abstract](300) +[HTML](1) +[PDF](578.5KB)
Abstract:

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

2017, 9(3): 335-390 doi: 10.3934/jgm.2017014 +[Abstract](473) +[HTML](9) +[PDF](857.5KB)
Abstract:

The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher-Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry--the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices.

Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the \begin{document}$QR$\end{document}, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples.

The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher-Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.

2017, 9(3): 391-410 doi: 10.3934/jgm.2017015 +[Abstract](302) +[HTML](2) +[PDF](1136.8KB)
Abstract:

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hörmander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

2017  Impact Factor: 0.561