Alignment of vector fields on manifolds via contraction mappings
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 mani...
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Kazan Federal University
2018-06-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/alignment-of-vector-fields-on-manifolds-via-403645.html |
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| author | O.N. Kachan Yu.A. Yanovich E.N. Abramov |
| author_facet | O.N. Kachan Yu.A. Yanovich E.N. Abramov |
| author_sort | O.N. Kachan |
| collection | DOAJ |
| description | 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) has been considered in the paper. One of the GSE subproblems is tangent space alignment. The original solution to this problem has been formulated as a generalized eigenvalue problem. In this formulation, it is plagued with numerical instability, resulting in suboptimal solutions to the subproblem and manifold reconstruction problem in general.
We have proposed an iterative algorithm to directly solve the tangent spaces alignment problem. As a result, we have obtained a significant gain in algorithm efficiency and time complexity. We have compared the performance of our method on various model data sets to show that our solution is on par with the approach to vector fields alignment formulated as an optimization on the Stiefel group. |
| format | Article |
| id | doaj-art-e8b13c02e6db4859b527f8be764eb052 |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2018-06-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-e8b13c02e6db4859b527f8be764eb0522025-01-03T00:06:19ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982018-06-011602300308Alignment of vector fields on manifolds via contraction mappingsO.N. Kachan0Yu.A. Yanovich1E.N. Abramov2Skolkovo Institute of Science and Technology, Moscow, 143026 RussiaSkolkovo Institute of Science and Technology, Moscow, 143026 Russia; Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, 127051 Russia; National Research University Higher School of Economics, Moscow, 101000 RussiaNational Research University Higher School of Economics, Moscow, 101000 RussiaAccording 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) has been considered in the paper. One of the GSE subproblems is tangent space alignment. The original solution to this problem has been formulated as a generalized eigenvalue problem. In this formulation, it is plagued with numerical instability, resulting in suboptimal solutions to the subproblem and manifold reconstruction problem in general. We have proposed an iterative algorithm to directly solve the tangent spaces alignment problem. As a result, we have obtained a significant gain in algorithm efficiency and time complexity. We have compared the performance of our method on various model data sets to show that our solution is on par with the approach to vector fields alignment formulated as an optimization on the Stiefel group.https://kpfu.ru/alignment-of-vector-fields-on-manifolds-via-403645.htmlmanifold learningdimensionality reductionnumerical optimizationvector field estimation |
| spellingShingle | O.N. Kachan Yu.A. Yanovich E.N. Abramov Alignment of vector fields on manifolds via contraction mappings Учёные записки Казанского университета: Серия Физико-математические науки manifold learning dimensionality reduction numerical optimization vector field estimation |
| title | Alignment of vector fields on manifolds via contraction mappings |
| title_full | Alignment of vector fields on manifolds via contraction mappings |
| title_fullStr | Alignment of vector fields on manifolds via contraction mappings |
| title_full_unstemmed | Alignment of vector fields on manifolds via contraction mappings |
| title_short | Alignment of vector fields on manifolds via contraction mappings |
| title_sort | alignment of vector fields on manifolds via contraction mappings |
| topic | manifold learning dimensionality reduction numerical optimization vector field estimation |
| url | https://kpfu.ru/alignment-of-vector-fields-on-manifolds-via-403645.html |
| work_keys_str_mv | AT onkachan alignmentofvectorfieldsonmanifoldsviacontractionmappings AT yuayanovich alignmentofvectorfieldsonmanifoldsviacontractionmappings AT enabramov alignmentofvectorfieldsonmanifoldsviacontractionmappings |