Optimization flows landing on the Stiefel manifold

Gao, Bin;Vary, Simon;Ablin, Pierre;Absil, Pierre-Antoine
(2022) 25th IFAC Symposium on Mathematical Theory of Networks and Systems MTNS 2022 — Location: Bayreuth, GERMANY (12.September.2022)

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Authors
  • Gao, BinUCLouvain
    Author
  • Vary, SimonUCLouvain
    Author
  • Ablin, PierreUniversite Paris-Dauphine, PSL University, France
    Author
  • Absil, Pierre-AntoineUCLouvain
    Author
Abstract
We study a continuous-time system that solves the optimization problem over the set of orthogonal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but eventually 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.
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Gao, B., Vary, S., Ablin, P., & Absil, P.-A. (2022). Optimization flows landing on the Stiefel manifold. Published. 25th IFAC Symposium on Mathematical Theory of Networks and Systems MTNS 2022, Bayreuth, GERMANY. https://doi.org/10.1016/j.ifacol.2022.11.023