Permutation-invariant log-Euclidean geometries on full-rank correlation matrices - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... (Preprint) Year :

Permutation-invariant log-Euclidean geometries on full-rank correlation matrices

Abstract

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 merge these two properties by defining the families of off-log metrics and log-scaled metrics. Firstly, we prove that the recently introduced off-log bijection is a diffeomorphism, allowing to pullback (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 secondly, we define the log-scaling 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.
Fichier principal
Vignette du fichier
SIMAX_YT_Permutation_invariant_log_Euclidean_correlation_HAL.pdf (409.28 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03878729 , version 1 (30-11-2022)

Identifiers

  • HAL Id : hal-03878729 , version 1

Cite

Yann Thanwerdas. Permutation-invariant log-Euclidean geometries on full-rank correlation matrices. 2022. ⟨hal-03878729⟩
33 View
75 Download

Share

Gmail Facebook Twitter LinkedIn More