Reparameterization invariant metric on the space of curves

This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0,1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM induces a first-order Sobolev metric on M with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .

Medians and means in Riemannian geometry: existence, uniqueness and computation

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Frechet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Frechet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.

Convex ordering for random vectors using predictable representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.