Equivariant log-extrinsic means on irreducible symmetric cones

Author

Chevallier, Emmanuel and Nielsen, Frank

Abstract

In this paper, we introduce a notion of mean on irreducible symmetric cones, based on the product decomposition between the determinant-one hypersurface and the determinant. Irreducible symmetric cones and their determinant one surfaces form an important class of spaces for statistics and data science, since they encompass positive definite self-adjoint matrices as well as Lorentz cones and hyperbolic spaces. By construction , log-extrinsic means have similar equivariance properties as those of Fréchet means. Moreover, the two means coincide under some symmetry assumption on the distribution. However, the log-extrinsic mean admits an explicit expression and is much simpler to compute. Numerical experiments show that the log-extrinsic means are a relevant alternative to log-Euclidean means. Furthermore, along with the log-extrinsic mean, we introduce a corresponding notion of Gaussian distributions, called log-extrinsic Gaussians. A classification experiment on stereo audio signals demonstrates the practical interest of the log-extrinsic framework.