StiefelLog swMATH ID: 42250 Software Authors: Nguyen, Du Description: Closed-form geodesics and optimization for Riemannian logarithms of Stiefel and flag manifolds. We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by Hüper, Markina and Silva Leite, reparameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank Stiefel manifolds. We follow the approach of minimizing the square Frobenius distance between a geodesic ending point to a given point on the manifold to compute the logarithm map and geodesic distance between two endpoints, using Fréchet derivatives to compute the gradient of this objective function. We focus on two optimization methods, extit{gradient descent} and L-BFGS. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. The method to translate directional derivatives using Fréchet derivatives to a gradient could potentially be applied to other matrix equations. Homepage: https://link.springer.com/article/10.1007/s10957-022-02012-3 Keywords: Stiefel manifold; geodesic; computer vision; flag manifold; logarithm map; Riemannian center of mass; Fréchet derivative Related Software: geomstats; GitHub; Python; minFunc; SciPy Cited in: 1 Publication Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Closed-form geodesics and optimization for Riemannian logarithms of Stiefel and flag manifolds. Zbl 1500.53056Nguyen, Du 2022 Cited by 1 Author 1 Nguyen, Du Dinh Cited in 1 Serial 1 Journal of Optimization Theory and Applications all top 5 Cited in 6 Fields 1 Algebraic geometry (14-XX) 1 Differential geometry (53-XX) 1 Manifolds and cell complexes (57-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Numerical analysis (65-XX) 1 Computer science (68-XX) Citations by Year