Rethinking the Riemannian Logarithm on Flag Manifolds as an Orthogonal Alignment Problem - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year :

Rethinking the Riemannian Logarithm on Flag Manifolds as an Orthogonal Alignment Problem

Abstract

Flags are sequences of nested linear subspaces of increasing dimension. They belong to smooth manifolds generalizing Grassmannians and bring a richer multi-scale point of view to the traditional subspace methods in statistical analysis. Hence, there is an increasing interest in generalizing the formulae and statistical methods already developed for Grassmannians to flag manifolds. In particular, it is critical to compute accurately and efficiently the geodesic distance and the logarithm due to their fundamental importance in geometric statistics. However, there is no explicit expression known in the case of flags. In this work, we exploit the homogeneous quotient space structure of flag manifolds and rethink the geodesic endpoint problem as an alignment of orthogonal matrices on their equivalence classes. The relaxed problem with the Frobenius metric surprisingly enjoys an explicit solution. This is the key to modify a previously proposed algorithm. We show that our explicit alignment step brings drastic improvements in accuracy, speed and radius of convergence, in addition to overcoming the combinatorial issues raised by the non-connectedness of the equivalence classes.
Fichier principal
Vignette du fichier
GSI-119.pdf (578.14 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-04100534 , version 1 (01-06-2023)

Licence

Attribution

Identifiers

  • HAL Id : hal-04100534 , version 1

Cite

Tom Szwagier, Xavier Pennec. Rethinking the Riemannian Logarithm on Flag Manifolds as an Orthogonal Alignment Problem. 2023. ⟨hal-04100534⟩
47 View
44 Download

Share

Gmail Facebook Twitter LinkedIn More