Alignment of vector fields on manifolds via contraction mappings

O.N. Kachan^a, Yu.A. Yanovich^a,b,c, E.N. Abramov^c

^aSkolkovo Institute of Science and Technology, Moscow, 143026 Russia

^bKharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, 127051 Russia

^cNational Research University Higher School of Economics, Moscow, 101000 Russia

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For citation: Kachan O.N., Yanovich Yu.A., Abramov E.N. Alignment of vector fields on manifolds via contraction mappings. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 300–308. 

Для цитирования: Kachan O.N., Yanovich Yu.A., Abramov E.N. Alignment of vector fields on manifolds via contraction mappings // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 300–308.

Abstract

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.

Keywords: manifold learning, dimensionality reduction, numerical optimization, vector field estimation

Acknowledgements. The study was supported by the Russian Science Foundation (project no. 14-50-00150).

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Received

October 11, 2017

   

Kachan Oleg Nikolaevich, PhD Student of the Center for Computational and Data-Intensive Science and Engineering

Skolkovo Institute of Science and Technology

ul. Nobelya, 3, Territory of the Innovation Center "Skolkovo'', Moscow, 143026 Russia

E-mail:  oleg.kachan@skoltech.ru

 

Yanovich Yury Alexandrovich, Candidate of Physical and Mathematical Sciences, Researcher of the Center for Computational and Data-Intensive Science and Engineering; Researcher of the Intelligent Data Analysis and Predictive Modeling Laboratory; Lecturer of the Faculty of Computer Science

Skolkovo Institute of Science and Technology

ul. Nobelya, 3, Territory of the Innovation Center ``Skolkovo'', Moscow, 143026 Russia

Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences

Bolshoy Karetny pereulok 19, str. 1, Moscow, 127051 Russia

National Research University "Higher School of Economics''

ul. Myasnitskaya, 20, Moscow, 101000 Russia

E-mail:  yury.yanovich@iitp.ru

 

Abramov Evgeny Nikolayevich, graduate student of Faculty of Computer Science

National Research University ``Higher School of Economics''

ul. Myasnitskaya, 20, Moscow, 101000 Russia

E-mail:  petzchner@gmail.com

 

 

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