The Journal of Geometric Mechanics: latest papers Latest articles for selected journal About simple variational splines from the Hamiltonian viewpoint Paula Balseiro, Teresinha J. Stuchi, Alejandro Cabrera and Jair Koiller Fri, 1 Sep 2017 08:00:00 GMT Probability measures on infinite-dimensional Stiefel 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. ]]>
Eleonora Bardelli and Andrea Carlo Giuseppe Mennucci Fri, 1 Sep 2017 08:00:00 GMT Complete spelling rules for the Monster tower over three-space Alex Castro, Wyatt Howard and Corey Shanbrom Fri, 1 Sep 2017 08:00:00 GMT Geometry of matrix decompositions seen through optimal transport and information geometry     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 $QR$, 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. ]]>
Klas Modin Fri, 1 Sep 2017 08:00:00 GMT Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry Stefan Sommer and Anne Marie Svane Fri, 1 Sep 2017 08:00:00 GMT