Geometrical and statistical properties of M-estimates of scatter on Grassmann manifolds

par Christian Mazza

jeudi 21 nov. 2019, 14:30 → 15:30 Europe/Paris

435 (ENS Lyon)

We consider data from the Grassmann manifold $G(m,r)$ of all vector subspaces of dimension $r$ of $m$ dimensional Euclidean space, and focus on the Grassmannian statistical model which is of common use in signal processing. Canonical Grassmannian distributions are indexed by parameters from the manifold $\mathcal{M}$ of positive definite symmetric matrices of determinant 1.  M-estimates of scatter (GE) for general probability measures $\mathcal{P}$ on $G(m,r)$ are studied. One of the novel features of this work is a strong use of the fact that $\mathcal{M}$ is a CAT(0) space with known visual boundary at infinity $\partial \mathcal{M}$.
We  recall that the sample space $G(m,r)$ is a part of $\partial \mathcal{M}$, and show
that the M-functional is a weighted Busemann function. 
We then consider  the almost sure convergence of (GE) when $\mathcal{P}$ is the empirical measure of a sample in $G(m,r)$ and provide a central limit theorem.
This talk is based on a joint work with Corina Ciobotaru.