Approximate joint diagonalization with Riemannian optimization on the general linear group

Florent Bouchard (1) , Bijan Afsari (2) , Jérôme Malick (3) , Marco Congedo (1)

1 GIPSA-VIBS - GIPSA - Vision and Brain Signal Processing
2 JHU - Johns Hopkins University
3 DAO - Données, Apprentissage et Optimisation

We consider the classical problem of approximate joint diagonalization of matrices, which can be cast as an optimization problem on the general linear group. We propose a versatile Riemannian optimization framework for solving this problem-unifiying existing methods and creating new ones. We use two standard Riemannian metrics (left-and right-invariant metrics) having opposite features regarding the structure of solutions and the model. We introduce the Riemannian optimization tools (gradient, retraction, vector transport) in this context, for the two standard non-degeneracy constraints (oblique and non-holonomic constraints). We also develop tools beyond the classical Riemannian optimization framework to handle the non-Riemannian quotient manifold induced by the non-holonomic constraint with the right-invariant metric. We illustrate our theoretical developments with numerical experiments on both simulated data and a real electroencephalographic recording.

Mots clés: Riemannian optimization; approximate joint diagonalization; oblique constraint; non-holonomic constraint

Domaines: Optimisation et contrôle [math.OC]; Traitement du signal et de l'image [eess.SP]

Florent Bouchard, Bijan Afsari, Jérôme Malick, Marco Congedo. Approximate joint diagonalization with Riemannian optimization on the general linear group. SIAM Journal on Matrix Analysis and Applications, 2020, 41 (1), pp.152-170. ⟨10.1137/18M1232838⟩. ⟨hal-02328480⟩