Paper ID sheet UCL-INMA-2022.03

Title

Optimization flows landing on the Stiefel manifold

Authors

Bin Gao, Simon Vary, Pierre Ablin, P.-A. Absil

Abstract

We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.

Key words

Stiefel manifold; Landing flow; Canonical metric; Riemannian gradient; Asymptotic stability

Status

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• Publisher's version: doi:10.1016/j.ifacol.2022.11.023
• Preprint

BibTeX entry

@article{GaoVaryAblinAbsil2022,
title = {Optimization flows landing on the Stiefel manifold⋆},
journal = {IFAC-PapersOnLine},
volume = {55},
number = {30},
pages = {25-30},
year = {2022},
note = {25th IFAC Symposium on Mathematical Theory of Networks and Systems MTNS 2022},
issn = {2405-8963},
doi = {https://doi.org/10.1016/j.ifacol.2022.11.023},
url = {https://www.sciencedirect.com/science/article/pii/S2405896322026519},
author = {Bin Gao and Simon Vary and Pierre Ablin and P.-A. Absil},
keywords = {},
abstract = {}
}