A.P. Kuleshov^a, A.V. Bernstein^a,b, Yu.A. Yanovich^a,b,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: Kuleshov A.P., Bernstein A.V., Yanovich Yu.A. Manifold learning based on kernel density estimation. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 327–338.

Для цитирования: Kuleshov A.P., Bernstein A.V., Yanovich Yu.A. Manifold learning based on kernel density estimation // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. – 2018. – Т. 160, кн. 2. – С. 327–338. 

Abstract

The problem of unknown high-dimensional density estimation has been considered. It has been suggested that the support of its measure is a low-dimensional data manifold. This problem arises in many data mining tasks. The paper proposes a new geometrically motivated solution to the problem in the framework of manifold learning, including estimation of an unknown support of the density.

Firstly, the problem of tangent bundle manifold learning has been solved, which resulted in the transformation of high-dimensional data into their low-dimensional features and estimation of the Riemann tensor on the data manifold. Following that, an unknown density of the constructed features has been estimated with the use of the appropriate kernel approach. Finally, using the estimated Riemann tensor, the final estimator of the initial density has been constructed.

Keywords: dimensionality reduction, manifold learning, manifold valued data, density estimation on manifold

Acknowledgements. The study by A.V. Bernstein and Yu.A. Yanovich was supported by the Russian Science Foundation (project no. 14-50-00150).

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Received

October 17, 2017

   

Kuleshov Alexander Petrovich, Doctor of Technical Sciences, Professor, Academician of the Russian Academy of Sciences, Rector

Skolkovo Institute of Science and Technology

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

E-mail:  kuleshov@skoltech.ru

 

Bernstein Alexander Vladimirovich, Doctor of Physical and Mathematical Sciences, Professor of the Center for Computational and Data-Intensive Science and Engineering; Leading Researcher of the Intelligent Data Analysis and Predictive Modeling Laboratory

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

E-mail:  a.bernstein@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

 

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