Geometrical and statistical properties of M-estimates of scatter on Grassmann manifolds
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We consider data from the Grassmann manifold G(m,r) of all vector subspaces of dimension r of R^m, and focus on the Grassmannian statistical model which is of common use in signal processing and statistics. Canonical Grassmannian distributions G_Σ on G(m,r) are indexed by parameters Σ from the manifold M= Pos_sym^1(m) of positive definite symmetric matrices of determinant 1. Robust M-estimates of scatter (GE) for general probability measures P on G(m,r) are studied. Such estimators are defined to be the maximizers of the Grassmannian log-likelihood -ℓ_P(Σ) as function of Σ. One of the novel features of this work is a strong use of the fact that M is a CAT(0) space with known visual boundary at infinity ∂M. We also recall that the sample space G(m,r) is a part of ∂M, show the distributions G_Σ are SL(m,R)--quasi-invariant, and that ℓ_P(Σ) is a weighted Busemann function. Let P_n =(δ_U_1+...+δ_U_n)/n be the empirical probability measure for n-samples of random i.i.d. subspaces U_i∈ G(m,r) of common distribution P, whose support spans R^m. For Σ_n and Σ_P the GEs of P_n and P, we show the almost sure convergence of Σ_n towards Σ as n→∞ using methods from geometry, and provide a central limit theorem for the rescaled process C_n = m/tr(Σ_P^-1Σ_n)g^-1Σ_n g^-1, where Σ =gg with g∈ SL(m,R) the unique symmetric positive-definite square root of Σ.
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