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Of all publications in the section: 6
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Article
Kachan O., Yanovich Y., Abramov E. Ученые записки Казанского университета. Серия: Физико-математические науки. 2018. Vol. 160. No. 2. P. 300-308.

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.

Added: Oct 29, 2020
Article
Abramov E., Yanovich Y. 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.

Added: Oct 29, 2020
Article
Kuleshov A. P., Bernstein A. V., Yanovich Y. Ученые записки Казанского университета. Серия: Физико-математические науки. 2018. Vol. 160. No. 2. P. 327-338.

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.

Added: Oct 28, 2020
Article
Kuleshov A. P., Bernstein A., Yanovich Y. Ученые записки Казанского университета. Серия: Физико-математические науки. 2018. Vol. 160. No. 2. P. 327-338.

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.

Added: Oct 28, 2020
Article
Koldanov P. Ученые записки Казанского университета. Серия: Физико-математические науки. 2018. Vol. 160. No. 2. P. 317-326.

Identification of network structures using the finite-size sample has been considered.

The concepts of random variables network and network model, which is a complete weighted

graph, have been introduced. Two types of network structures have been investigated: network

structures with an arbitrary number of elements and network structures with a fixed number

of elements of the network model. The problem of identification of network structures has

been investigated as a multiple testing problem. The risk function of statistical procedures for

identification of network structures can be represented as a linear combination of expected

numbers of incorrectly included elements and incorrectly non-included elements. The sufficient

conditions of optimality for statistical procedures for network structures identification with

an arbitrary number of elements have been given. The concept of statistical uncertainty of

statistical procedures for identification of network structures has been introduced.

Added: Feb 13, 2019
Article
Сысоева Л. Н. Ученые записки Казанского университета. Серия: Физико-математические науки. 2014. Т. 156. № 3. С. 116-122.

In this paper, we consider the problem of implementation of Boolean functions by generalized alpha-formulas. The notion of generalized alpha-formula is introduced. For a given set of Boolean functions, we define the notion of a universal set of generalized alpha-formulas. We also propose the notion of dual generalized alpha-formulas and formulate the principle of duality for generalized alpha-formulas. The presence of universal sets of generalized alpha-formulas is proved for every n ≥ 2 for the sets of 0-preserving and 1-preserving Boolean functions of the variables x_1, x_2, \dots, x_n.

Added: Nov 11, 2017