Marc Arnaudon

Riemannian Medians and Means With Applications to Radar Signal Processing

We develop a new geometric approach for high resolution Doppler processing based on the Riemannian geometry of Toeplitz covariance matrices and the notion of Riemannian p -means. This paper summarizes briefly our recent work in this direction. First of all, we introduce radar data and the problem of target detection. Then we show how to transform the original radar data into Toeplitz covariance matrices. After that, we give our results on the Riemannian geometry of Toeplitz covariance matrices. In order to compute p-means in practical cases, we propose deterministic and stochastic algorithms, of which the convergence results are given, as well as the rate of convergence and error estimates. Finally, we propose a new detector based on Riemannian median and show its advantage over the existing processing methods.

Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions

In this paper we generalize the continuous-discrete extended Kalman filter (CD-EKF) to the case where the state and the observations evolve on connected unimodular matrix Lie groups. We propose a new assumed density filter called continuous-discrete extended Kalman filter on Lie groups (CD-LG-EKF). It is built upon a geometrically meaningful modeling of the concentrated Gaussian distribution on Lie groups. Such a distribution is parametrized by a mean and a covariance matrix defined on the Lie group and in its associated Lie algebra respectively. Our formalism yields tractable equations for both non-linear continuous time propagation and discrete update of the distribution parameters under the assumption that the posterior distribution of the state is a concentrated Gaussian. As a side effect, we contribute to the derivation of the first and second order differential of the matrix Lie group logarithm using left connection. We also show that the CD-LG-EKF reduces to the usual CD-EKF if the state and the observations evolve on Euclidean spaces. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose filtering problem with observations on Lie group. In this application, the CD-LG-EKF significantly outperforms two constrained non-linear filters (one based on a linearization technique and the other on the unscented transform) applied on the embedding space of the Lie group.

COMPLETE LIFTS OF CONNECTIONS AND STOCHASTIC JACOBI FIELDS

Differentiable families of V-martingales on manifolds are investigated: their infinitesimal variation provides a notion of stochastic Jacobi fields. Such objects are known (2) to be martingales taking values in the tangent bundle when the latter is equipped with the complete lift of the connection V. We discuss various characterizations of TM-valued martingales. When applied to specific families of V-martingales which appear in connection with the heat flow for maps between Riemannian manifolds, our results allow to establish formulas giving a stochastic representation for the differential of solutions to the nonlinear heat equation. As an application, we prove local and global gradient estimates for harmonic maps of bounded dilatation. 0 Elsevier, Paris

Stochastic Euler-Poincaré reduction

We prove a Euler-Poincare reduction theorem for stochastic processes taking values on a Lie group, which is a generalization of the reduction argument for the deterministic case [J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, (Texts in Applied Mathematics). (Springer, 2003)]. We also show examples of its application to SO(3) and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

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.

On approximating the Riemannian 1-center

We generalize the Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) [6] to arbitrary Riemannian geometries, and study the corresponding convergence rate. We then show how to instantiate this generic algorithm to two particular settings: (1) the hyperbolic geometry, and (2) the Riemannian manifold of symmetric positive definite matrices.

Geometry of Covariance Matrices and Computation of Median

In this paper, we consider the manifold of covariance matrices of order n parametrized by reflection coefficients which are derived from Levinson’s recursion of autoregressive model. The explicit expression of the reparametrization and its inverse are obtained. With the Riemannian metric given by the Hessian of a Kahler potential, we show that the manifold is in fact a Cartan‐Hadamard manifold with lower sectional curvature bound ‐4. The explicit expressions of geodesics are also obtained. After that we introduce the notion of Riemannian median of points lying on a Riemannian manifold and give a simple algorithm to compute it. Finally, some simulation examples are given to illustrate the applications of the median method to radar signal processing.

Horizontal martingales in vector bundles

1. Introduction and Notations 2. Complete Lifts to Tangent Bundles 3. Deformed Antidevelopment 4. Horizontal Lifts to Vector Bundles 5. A General Class of Lifts to Vector Bundles 6. Complete Lifts to Cotangent Bundles 7. Complete Lifts to Exterior Bundles 8. Complete Lifts to Dirac Bundles 9. Martingales in the Tangent Space Related to Harmonic Maps 10. Martingales in the Exterior Tangent Bundle Related to Harmonic Forms References

Horizontal Diffusion in C 1 Path Space

We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing un- der the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge- Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This con- struction provides an alternative method to filtering out redundant noise.

A probabilistic approach to the Yang-Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇0U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U −1 ∇0U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U −1 ∇0U is given, both in terms of change of time and in terms of scaling of U −1 ∇0U . This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.  2002 Editions scientifiques et medicales Elsevier SAS.