Non Positively Curved Metric in the Space of Positive Definite Infinite Matrices
نویسنده
چکیده
We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. Moreover, these spaces of finite matrices are naturally imbedded in Σ∞.
منابع مشابه
Learning with Iwasawa Coordinates
Finding a good metric over the input space plays a fundamental role in machine learning. Most existing techniques assume the Mahalanobis metric without incorporating the geometry of Pn, the space of n×n symmetric positive-definite (SPD) matrices, which leads to difficulties in the optimization procedure used to learn the metric. In this paper, we introduce a novel algorithm to learn the Mahalan...
متن کاملGyrovector Spaces on the Open Convex Cone of Positive Definite Matrices
In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces, which are the Einstein and M"{o}bius gyrovector spaces. We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its...
متن کاملHoroball Hulls and Extents in Positive Definite Space
The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometric structures and algorithms for analyzing collections of such matrices. A key technical contribu...
متن کاملNon-linear ergodic theorems in complete non-positive curvature metric spaces
Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...
متن کاملLog-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces
This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance o...
متن کامل