A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints

Nicolas Boumal 1, *
* Auteur correspondant
1 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : We propose a new algorithm to solve optimization problems of the form min f (X) for a smooth function f under the constraints that X is positive semidefinite and the diagonal blocks of X are small identity matrices. Such problems often arise as the result of relaxing a rank constraint (lifting). In particular, many estimation tasks involving phases, rotations, orthonormal bases or permutations fit in this framework, and so do certain relaxations of combinatorial problems such as Max-Cut. The proposed algorithm exploits the facts that (1) such formulations admit low-rank solutions, and (2) their rank-restricted versions are smooth optimization problems on a Riemannian manifold. Combining insights from both the Riemannian and the convex geometries of the problem, we characterize when second-order critical points of the smooth problem reveal KKT points of the semidefinite problem. We compare against state of the art, mature software and find that, on certain interesting problem instances, what we call the staircase method is orders of magnitude faster, is more accurate and scales better. Code is available.
Type de document :
Pré-publication, Document de travail
2015
Liste complète des métadonnées

Littérature citée [82 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01213337
Contributeur : Nicolas Boumal <>
Soumis le : jeudi 8 octobre 2015 - 12:07:19
Dernière modification le : mardi 24 avril 2018 - 17:20:14
Document(s) archivé(s) le : samedi 9 janvier 2016 - 10:21:01

Fichier

1506.00575v1.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01213337, version 1

Collections

Citation

Nicolas Boumal. A Riemannian low-rank method for optimization over semidefinite matrices with block-diagonal constraints. 2015. 〈hal-01213337〉

Partager

Métriques

Consultations de la notice

222

Téléchargements de fichiers

247