Abstract
Explicit matrix-type formulas for gradients and Hessians of smooth functions on the compact real Stiefel manifold with respect to a whole class of (pseudo-)Riemannian metrics are presented. This includes explicit formulas for corresponding normal spaces and associated orthogonal projections. It turns out that some well-known formulas are reproduced, moreover, it is shown that they are even valid in a much bigger context. All proofs are included, some of them for the first time. A numerical experiment is added as well.
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References
Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008). https://doi.org/10.1515/9781400830244
Eaton, J.W., Bateman, D., Hauberg, S., Wehbring, R.: GNU Octave version 6.3.0 manual: a high-level interactive language for numerical computations (2021). https://www.gnu.org/software/octave/doc/v6.3.0/
Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1999). https://doi.org/10.1137/S0895479895290954
Gallier, J., Quaintance, J.: Differential Geometry and Lie Groups–A Computational Perspective. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-46040-2
Gallier, J., Quaintance, J.: Differential Geometry and Lie Groups–A Second Course. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-46047-1
Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Springer, London (1994). https://doi.org/10.1007/978-1-4471-3467-1
Hüper, K., Markina, I., Silva Leite, F.: A Lagrangian approach to extremal curves on Stiefel manifolds. J. Geom. Mech. 13(1), 55–72 (2021). https://doi.org/10.3934/jgm.2020031
Nguyen, D.: Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds (2021). https://arxiv.org/abs/2103.13327
Nguyen, D.: Operator-valued formulas for Riemannian gradient and Hessian and families of tractable metrics (2021). https://arxiv.org/abs/2009.10159
O’Neill, B.: Semi-Riemannian Geometry. Academic Press Inc., New York (1983)
Smith, S.T.: Geometric optimization methods for adaptive filtering. Ph.D. thesis, Harvard University, Cambridge (1993)
Ye, K., Wong, K.S.W., Lim, L.H.: Optimization on flag manifolds (2019). https://arxiv.org/abs/1907.00949
Acknowledgements
This work has been supported by the German Federal Ministry of Education and Research (BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020 - DyCA).
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Schlarb, M., Hüper, K. (2022). Optimization on Stiefel Manifolds. In: Brito Palma, L., Neves-Silva, R., Gomes, L. (eds) CONTROLO 2022. CONTROLO 2022. Lecture Notes in Electrical Engineering, vol 930. Springer, Cham. https://doi.org/10.1007/978-3-031-10047-5_32
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DOI: https://doi.org/10.1007/978-3-031-10047-5_32
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