PERMUTATION-INVARIANT LOG-EUCLIDEAN GEOMETRIES ON FULL-RANK CORRELATION MATRICES

There is a growing interest in defining specific tools on correlation matrices which depart from those suited to SPD matrices. Several geometries have been defined on the open elliptope of full-rank correlation matrices: some are permutation-invariant, some others are log-Euclidean, i.e., diffeomorphic to a Euclidean space. In this work, we prove the existence of permutation-invariant log-Euclidean metrics by defining the families of off-log metrics and log-scaled metrics. First, we prove that the recently introduced off-log bijection is a smooth diffeomorphism, allowing pullback of (permutation-invariant) inner products. We introduce the ``cor-inverse"" involution on the open elliptope, which can be seen as analogous to the inversion of SPD matrices. We show that off-log metrics are not inverse-consistent. That is why, second, we define the log-scaling smooth diffeomorphism between the open elliptope and the vector space of symmetric matrices with null row sums. This map is based on the congruence action of positive diagonal matrices on SPD matrices, more precisely on the existence and uniqueness of a ``scaling,"" i.e., an SPD matrix with unit row sums within an orbit. Thanks to this multiplicative approach, log-scaled metrics are inverse-consistent. We provide the main Riemannian operations in closed form for the two families modulo the computation of the respective bijections.