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Petr Chunaev

Publications and source records attributed to Petr Chunaev.

12 recordsLinked to original sources
arXiv2022

Synthetic Data-Based Simulators for Recommender Systems: A Survey

This survey aims at providing a comprehensive overview of the recent trends in the field of modeling and simulation (M&S) of interactions between users and recommender systems and applications of the M&S to the performance improvement of industrial recommender engines. We start with the motivation behind the development of frameworks implementing the simulations -- simulators -- and the usage of them for training and testing recommender systems of different types (including Reinforcement Learning ones). Furthermore, we provide a new consistent classification of existing simulators based on their functionality, approbation, and industrial effectiveness and moreover make a summary of the simulators found in the research literature. Besides other things, we discuss the building blocks of simulators: methods for synthetic data (user, item, user-item responses) generation, methods for what-if experimental analysis, methods and datasets used for simulation quality evaluation (including the methods that monitor and/or close possible simulation-to-reality gaps), and methods for summarization of experimental simulation results. Finally, this survey considers emerging topics and open problems in the field.

cs.IR↗
arXiv2019

Community detection in node-attributed social networks: a survey

Community detection is a fundamental problem in social network analysis consisting in unsupervised dividing social actors (nodes in a social graph) with certain social connections (edges in a social graph) into densely knitted and highly related groups with each group well separated from the others. Classical approaches for community detection usually deal only with network structure and ignore features of its nodes (called node attributes), although many real-world social networks provide additional actors' information such as interests. It is believed that the attributes may clarify and enrich the knowledge about the actors and give sense to the communities. This belief has motivated the progress in developing community detection methods that use both the structure and the attributes of network (i.e. deal with a node-attributed graph) to yield more informative and qualitative results. During the last decade many such methods based on different ideas have appeared. Although there exist partial overviews of them, a recent survey is a necessity as the growing number of the methods may cause repetitions in methodology and uncertainty in practice. In this paper we aim at describing and clarifying the overall situation in the field of community detection in node-attributed social networks. Namely, we perform an exhaustive search of known methods and propose a classification of them based on when and how structure and attributes are fused. We not only give a description of each class but also provide general technical ideas behind each method in the class. Furthermore, we pay attention to available information which methods outperform others and which datasets and quality measures are used for their evaluation. Basing on the information collected, we make conclusions on the current state of the field and disclose several problems that seem important to be resolved in future.

cs.SI↗
arXiv2019

Gorin's problem for individual simple partial fractions

The main result of the paper is a lower estimate for the moduli of imaginary parts of the poles of a simple partial fraction (i.e. the logarithmic derivative of an algebraic polynomial) under the condition that the $L^\infty(\mathbb{R})$-norm of the fraction is unit (Gorin's problem). In contrast to the preceding results, the estimate takes into account the residues associated with the poles. Moreover, a new estimate for the moduli is obtained in the case when the $L^\infty(\mathbb{R})$-norm of the derivative of the simple partial fraction is unit (Gelfond's problem).

math.CA↗
arXiv2019

Interpolation by generalized exponential sums with equal weights

Here we solve Pad\'e and Prony interpolation problems for the generalized exponential sums with equal weights: $$H_n(z; h)=\frac{\mu}{n}\sum_{k=1}^n h(\lambda_k z),\quad \text{where}\quad \mu,\lambda_k\in \mathbb{C},$$ and $h$ is a fixed analytic function under few natural assumptions. The interpolation of a function $f$ by $H_n$ is due to properly chosen $\mu$ and $\{\lambda_k\}_{k=1}^n$, which depend on $f$, $h$ and $n$. The sums $H_n$ are related to the $h$-sums and generalized exponential sums, i.e. to $$\mathcal{H}^*_n(z; h)=\sum_{k=1}^n \lambda_k h(\lambda_k z)\quad \text{and}\quad\mathcal{H}_n(z; h):=\sum_{k=1}^n \mu_k h(\lambda_k z),\quad \text{where}\quad \mu_k,\lambda_k\in \mathbb{C},$$ which generalize many classical approximants and whose properties are actively studied. As for the Pad\'e problem, we show that $H_n$ and $\mathcal{H}_n^*$ have similar constructions and rates of interpolation, whereas calculating $H_n$ requires less arithmetic operations. Although the Pad\'e problem for $\mathcal{H}_n$ is known to have a doubled interpolation rate with respect to $\mathcal{H}_n^*$ and thus to $H_n$, it can be however unsolvable in many useful cases and this may entirely eliminate the advantage of $\mathcal{H}_n$. We show that, in contrast to $\mathcal{H}_n$, the Pad\'e problem for $H_n$ always has a unique solution. More importantly, we also obtain efficient estimates for $\mu$ and $\lambda_k$, valuable by themselves, and use them in further evaluating interpolation quality and in applications. The Pad\'e problem and estimates provide a basis for managing the more interesting Prony problem for exponential sums with equal weights $H_n(z;\exp)$, i.e. when $h(z)=\exp(z)$. We show that it is uniquely solvable and surprisingly $\mu$ and $\lambda_k$ can be efficiently estimated. This is in sharp contrast to the case of well-known exponential sums $\mathcal{H}_n(z;\exp)$.

math.CA↗
arXiv2018

A family of singular integral operators which control the Cauchy transform

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of $T_{k_t}$ with a fixed $t\in (-t_0,0)$, where $t_0>0$ is an absolute constant, implies that $E$ is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the $L^2$-boundedness of the Cauchy transform, which is the key ingredient for the bi-Lipschitz invariance of analytic capacity.

math.CA↗
arXiv2016

Quadrature formulas with variable nodes and Jackson-Nikolskii inequalities for rational functions

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our $(\infty,2)$- and $(\infty,4)$-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors.

math.CA↗
arXiv2016

Singular integrals unsuitable for the curvature method whose $L^2$-boundedness still implies rectifiability

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(μ)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $μ$, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the $L^2(μ)$-boundedness of the operator still implies that the support of $μ$ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors-David regularity conditions.

math.CA↗
arXiv2016

Approximation by amplitude and frequency operators

We study Padé interpolation at the node $z=0$ of functions $f(z)=\sum_{m=0}^{\infty} f_m z^m$, analytic in a neighbourhood of this node, by amplitude and frequency operators (sums) of the form $$ \sum_{k=1}^n μ_k h(λ_k z), \qquad μ_k,λ_k\in \mathbb{C}. $$ Here $h(z)=\sum_{m=0}^{\infty} h_m z^m$, $h_m\ne 0$, is a fixed (basis) function, analytic at the origin, and the interpolation is carried out by an appropriate choice of amplitudes $μ_k $ and frequencies $λ_k$. The solvability of the $2n$-multiple interpolation problem is determined by the solvability of the associated moment problem $$ \sum_{k=1}^nμ_k λ_k^m={f_m}/{h_m}, \qquad m=\overline{0,2n-1}. $$ In a number of cases, when the moment problem is consistent, it can be solved by the classical method due to Prony and Sylvester, moreover, one can easily construct the corresponding interpolating sum too. In the case of inconsistent moment problems, we propose a regularization method, which consists in adding a special binomial $c_1z^{n-1}+c_2 z^{2n-1}$ to an amplitude and frequency sum so that the moment problem, associated with the sum obtained, can be already solved by the method of Prony and Sylvester. This approach enables us to obtain interpolation formulas with $n$ nodes $λ_k z$, being exact for the polynomials of degree $\le 2n-1$, whilst traditional formulas with the same number of nodes are usually exact only for the polynomials of degree $\le n-1$. The regularization method is applied to numerical differentiation and extrapolation.

math.CA↗
arXiv2016

A new family of singular integral operators whose $L^2$-boundedness implies rectifiability

Let $E \subset \mathbb{C}$ be a Borel set such that $0<\mathcal{H}^1(E)<\infty$. David and L\'eger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\, z/|z|^2$, $z\in \mathbb{C}\setminus\{0\}$) has the following property $(*)$: the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, $n\in \mathbb{N}$. In this paper, we prove that the property $(*)$ is valid for operators associated to the much wider class of kernels $(\textrm{Re}\, z)^{2N-1}/|z|^{2N}+t\cdot(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, where $n,N$ are positive integer numbers such that $N\ge n$, and $t\in \mathbb{R}\setminus (t_1,t_2)$ with $t_1,t_2$ depending only on $n$ and $N$.

math.CA↗
arXiv2014

On Hölder and Minkowski Type Inequalities

We obtain inequalities of Hölder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of non-negative weights.

math.CA↗
arXiv2014

Logarithmic Derivatives of Least Deviation from Zero

We study least deviation of logarithmic derivatives of real-valued polynomials with a fixed root from zero on the segment $[-1;1]$ in the uniform norm with the weight $\sqrt{1-x^2}$ and without it. Basing on results of Komarov and Novak and on a certain determinant identity due to Borchardt, we also establish a criterion for best uniform approximation of continuous real-valued functions by logarithmic derivatives in terms of a Chebyshev alternance.

math.CA↗