Estimation Of Smooth Vector Fields On Manifolds By Optimization On Stiefel Group

E. Abramov; Y. Yanovich

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Estimation Of Smooth Vector Fields On Manifolds By Optimization On Stiefel Group

Ученые записки Казанского университета. Серия: Физико-математические науки. 2018. Vol. 160. No. 2. P. 220–228.

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.

Language: English

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Keywords: Manifold Learning Dimensionality reduction vector field estimation optimization on Stiefel manifold

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According to the manifold hypothesis, high-dimensional data can be viewed and meaning- fully represented as a lower-dimensional manifold embedded in a higher dimensional feature space. Manifold learning is a part of machine learning where an intrinsic data representation is uncovered based on the manifold hypothesis. Many manifold learning algorithms were developed. The one called Grassmann&Stiefel eigenmaps ...

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Alignment Of Vector Fields On Manifolds Via Contraction Mappings

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According to the manifold hypothesis, high-dimensional data can be viewed and meaningfully represented as a lower-dimensional manifold embedded in a higher dimensional feature space. Manifold learning is a part of machine learning where an intrinsic data representation is uncovered based on the manifold hypothesis. Many manifold learning algorithms were developed. The one called Grassmann & Stiefel eigenmaps (GSE) ...

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