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Vol 80, No 6 (2025)

Year: 2025
Articles: 13
URL: https://journals.rcsi.science/0042-1316/issue/view/24436

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Regularity of solutions to Fokker–Planck–Kolmogorov equations

Bogachev V.I., Shaposhnikov S.V.

Abstract

A survey of the regularity properties of solutions to Fokker-Planck-Kolmogorov equations of elliptic and parabolic type is presented. Conditions for the existence of the densities of solutions and some local properties of such densities, such as boundedness, continuity, Hölder continuity, and Sobolev continuity are discussed, as well as some global properties such as estimates on the whole space, high integrability, and membership in Sobolev classes in the whole space. New results on properties of solutions in the case of low regularity of coefficients are also presented.

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Uspekhi Matematicheskikh Nauk. 2025;80(6):3-44

3-44

(Rus)

(JATS XML)

Convergence of a multilayer perceptron to histogram Bayesian regression

Eliseev N.A., Perminov A.I., Turdakov D.Y.

Abstract

The problem of enhancing the interpretability and consistency of Baysesian classifier solutions in approximating the empirical data by means of a multilayer perceptron is under consideration. Histogram regression preserves transparency and statistical interpretation but is limited by memory requirements ($O(n)$) and weak scalability, while a multilayer perceptron provides a memory efficient representation ($O(1)$)and high computational efficiency in combination with limited interpretability. The focus is on a unary learning scheme, when the training sample consists of examples in the same target class and additional background points which are uniformly distributed over a compact subset of the feature space. This approach enables one to treat each class separately and implement the failure mechanism outside the data support, which enhances the model reliability. It is proposed to consider the perceptron output as a consistent analogue of the histogram class interval induced by the linearity cells of the perceptron. It is proved that under the natural assumptions of regularity and controlled growth of architecture the output function of a multilayer perseptron is consistent and equivalent to a histogram estimator. Theoretical consistency is rigorously ðroved in the case of a fixed first layer, while numerical experiments confirm the applicability of the results to models all of whose layers are trained. Thus histogram interpretation ensures the statistical verification of the consistency of perceptron approximation and addscredibility to classification solutions in the framework of a unary model.

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Uspekhi Matematicheskikh Nauk. 2025;80(6):45-72

45-72

(Rus)

(JATS XML)

Generalized chord diagrams and weight systems

Kazarian M.E., Krasil'nikov E.S., Lando S.K., Shapiro M.Z., Zaitsev M.R.

Abstract

The paper is devoted to a description of the recent progress in understanding the extension of Lie algebra weight systems to permutations. Lie algebra weight systems are functions on chord diagrams arising naturally in Vassiliev's theory of finite-type knot invariants. These functions satisfy certain linear restrictions known as Vassiliev's 4-term relations. Chord diagrams can be interpreted as fixed-point-free involutions in symmetric groups, and an extension of Lie algebra weight systems to arbitrary permutations was aimed at finding an efficient way to compute their values. We show that this extension is of interest on its own, which suggests introducing the notion of weight system on permutations. To this end we define generalized Vassiliev's relations for permutations, which reduce to conventional ones for chord diagrams. We also describe the corresponding Hopf algebra structures on spaces of permutations that match the classical Hopf algebra structure on the space of chord diagrams modulo 4-term relations. Among main results of the paper is an explicit formula for the average value of the universal $\mathfrak{gl}$-weight system on permutations. This formula implies, in particular, that this average value is a tau-function for the Kadomtsev-Petviashvili hierarchy of partial differential equations. Its proof is based on an analysis of a quantum version of the universal $\mathfrak{gl}$-weight system.

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Uspekhi Matematicheskikh Nauk. 2025;80(6):73-136

73-136

(Eng)

(JATS XML)

Accelerated Bregman gradient methods for relatively smooth and relatively Lipschitz continuous minimization problems

Savchuk O.S., Alkousa M.S., Shushko A.I., Vyguzov A.A., Stonyakin F.S., Pasechnyuk D.A., Gasnikov A.V.

Abstract

We propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with inexact oracle. We consider the problem of minimizing a convex, differentiable, and relatively smooth function relative to a reference convex function. The first proposed method is based on a similar triangles method with inexact oracle, which uses a special triangular scaling property of the Bregman divergence used. The other proposed methods are non-adaptive and adaptive (tuning to the relative smoothness parameter) accelerated Bregman proximal gradient methods with inexact oracle. These methods are universal in the sense that they apply not only to relatively smooth but also to relatively Lipschitz continuous optimization problems. We also introduce an adaptive intermediate Bregman method, which interpolates between slower but more robust non-accelerated algorithms and faster but less robust accelerated algorithms. We conclude the paper with the results of numerical experiments demonstrating the advantages of the proposed algorithms for the Poisson inverse problem.

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Uspekhi Matematicheskikh Nauk. 2025;80(6):137-172

137-172