Speaker: Bruno Iannazzo Affiliation: Università degli studi di Perugia Venue: Aula Magna, Dipartimento di Matematica Time: Lunedì, 6/12/2021, 15:00

Title: New geometries on positive-definite matrices and their relation to the power means

We introduce a new family of geometries on the cone of positive definite matrices obtained from the Hessian of the power potential and provide explicit expressions for related quantities such as geodesics and distances. This generalizes in some sense the geometry obtained from the Hessian of the logarithmic potential, whose geodesic has been understood as the weighted geometric mean of two matrices. Indeed, the new geometries provide a definition of matrix power mean alternative to the existing ones. We discuss some properties of the proposed power mean and relate it with the geometric mean. Finally, we discuss on how these geometries can be used to accelerate the convergence of optimization algorithms for the matrix geometric mean of several variables.

Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4