Geodesic
Estimation of smooth vector fields on manifolds by optimization on Stiefel group
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 220-228

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Real data are usually characterized by high dimensionality. However, real data obtained from real sources, due to the presence of various dependencies between data points and limitations on their possible values, form, as a rule, form a small part of the high-dimensional space of observations. The most common model is based on the hypothesis that data lie on or near a manifold of a smaller dimension. This assumption is called the manifold hypothesis, and inference and calculations under it are called manifold learning. Grassmann Stiefel eigenmaps is a manifold learning algorithm. One of its subproblems has been considered in the paper: estimation of smooth vector fields by optimization on the Stiefel group. A two-step algorithm has been introduced to solve the problem. Numerical experiments with artificial data have been performed.
Keywords: manifold learning, dimensionality reduction, vector field estimation, optimization on Stiefel manifold. 
@article{UZKU_2018_160_2_a1,
     author = {E. N. Abramov and Yu. A. Yanovich},
     title = {Estimation of smooth vector fields on manifolds by optimization on {Stiefel} group},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {220--228},
     publisher = {mathdoc},
     volume = {160},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a1/}
}
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E. N. Abramov; Yu. A. Yanovich. Estimation of smooth vector fields on manifolds by optimization on Stiefel group. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 220-228. http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a1/