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Vugar Ismailov

Publications and source records attributed to Vugar Ismailov.

At least 19 recordsLinked to original sources
arXiv2026

Two fixed functions can approximate any continuous function using only addition and composition

We prove that two fixed univariate functions, namely, an arbitrary continuous non-affine function and a particular affine function, are sufficient to approximate continuous functions of one variable under the operations of addition and composition. The same fixed functions can also be used to approximate multivariate continuous functions, provided that the coordinate functions are also available. We also show that the number of generators can be reduced from two to one. We construct a specific continuous function that generates a dense class in the univariate setting and, together with the coordinate functions, in the multivariate setting.

math.FA↗
arXiv2026

Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

We analyze the universal approximation property of Kolmogorov-Arnold Networks (KANs) in terms of their edge functions. If these functions are all affine, then universality clearly fails. How many non-affine functions are needed, in addition to affine ones, to ensure universality? We show that a single one suffices. More precisely, we prove that deep KANs in which all edge functions are either affine or equal to a fixed continuous function $\sigma$ are dense in $C(K)$ for every compact set $K\subset\mathbb{R}^n$ if and only if $\sigma$ is non-affine. In contrast, for KANs with exactly two hidden layers, universality holds if and only if $\sigma$ is nonpolynomial. We further show that the full class of affine functions is not required; it can be replaced by a finite set without affecting universality. In particular, in the nonpolynomial case, a fixed family of five affine functions suffices when the depth is arbitrary. More generally, for every continuous non-affine function $\sigma$, there exists a finite affine family $A_\sigma$ such that deep KANs with edge functions in $A_\sigma\cup\{\sigma\}$ remain universal. We also prove that KANs with the spline-based edge parameterization introduced by Liu et al.~\cite{Liu2024} are universal approximators in the classical sense, even when the spline degree and knot sequence are fixed in advance.

cs.LG↗
arXiv2026

Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem

Deep Operator Networks (DeepONets) provide a branch-trunk neural architecture for approximating nonlinear operators acting between function spaces. In the classical operator approximation framework, the input is a function $u\in C(K_1)$ defined on a compact set $K_1$ (typically a compact subset of a Banach space), and the operator maps $u$ to an output function $G(u)\in C(K_2)$ defined on a compact Euclidean domain $K_2\subset\mathbb{R}^d$. In this paper, we develop a topological extension in which the operator input lies in an arbitrary Hausdorff locally convex space $X$. We construct topological feedforward neural networks on $X$ using continuous linear functionals from the dual space $X^*$ and introduce topological DeepONets whose branch component acts on $X$ through such linear measurements, while the trunk component acts on the Euclidean output domain. Our main theorem shows that continuous operators $G:V\to C(K;\mathbb{R}^m)$, where $V\subset X$ and $K\subset\mathbb{R}^d$ are compact, can be uniformly approximated by such topological DeepONets. This extends the classical Chen-Chen operator approximation theorem from spaces of continuous functions to locally convex spaces and yields a branch-trunk approximation theorem beyond the Banach-space setting.

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arXiv2026

Universality of shallow and deep neural networks on non-Euclidean spaces

We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We focus on the universal approximation property and establish general conditions under which such networks are dense in spaces of continuous vector-valued functions on arbitrary topological spaces and, in particular, locally convex spaces. Universality results obtained in the arbitrary-width case extend classical approximation theorems to non-Euclidean spaces. We also consider the deep narrow setting, in which the width of each hidden layer is uniformly bounded while the depth is allowed to grow. We identify conditions under which such networks retain the universal approximation property. As a concrete example, we employ Ostrand's extension of the Kolmogorov superposition theorem to derive an explicit universality result for products of compact metric spaces, with width bounds expressed in terms of topological dimension.

math.GN↗
arXiv2026

Universal Approximation Theorem for Input-Connected Multilayer Perceptrons

We present the Input-Connected Multilayer Perceptron (IC-MLP), a feedforward neural network architecture in which each hidden neuron receives, in addition to the outputs of the preceding layer, a direct affine connection from the raw input. We first study this architecture in the univariate setting and give an explicit and systematic description of IC-MLPs with an arbitrary finite number of hidden layers, including iterated formulas for the network functions. In this setting, we prove a universal approximation theorem showing that deep IC-MLPs can approximate any continuous function on a closed interval of the real line if and only if the activation function is nonlinear. We then extend the analysis to vector-valued inputs and establish a corresponding universal approximation theorem for continuous functions on compact subsets of $\mathbb{R}^n$.

cs.LG↗
arXiv2025

Metrics Over Merit: The Hidden Costs of Citation Impact in Research

Once upon a time, scientists' worth was measured by their ideas, proofs, and perhaps how eloquently they debated Hilbert's problems at seminars. But now, citation metrics have come to center stage and handed us new masters: FWCI and CNCI. This paper critically, and with a touch of satire, examines how these seemingly objective metrics are shaping, and often distorting, the scientific landscape. Through examples and analysis, we highlight the consequences of relying too heavily on such indicators in evaluating researchers and scientific contributions.

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arXiv2025

On shallow feedforward neural networks with inputs from a topological space

We study feedforward neural networks with inputs from a topological space (TFNNs). We prove a universal approximation theorem for shallow TFNNs, which demonstrates their capacity to approximate any continuous function defined on this topological space. As an application, we obtain an approximative version of Kolmogorov's superposition theorem for compact metric spaces.

cs.LG↗
arXiv2025

On the Kurepa and inhomogeneous Cauchy functional equations

It follows from de Bruijn's results that if a continuous or $k$-th order continuously differentiable function $F(x,y)$ is a solution of the Kurepa functional equation, then it can be expressed as $F(x,y)=f(x+y)-f(x)-f(y)$ with the continuous $f$ or the $k$-th order continuously differentiable $f$, respectively. These two facts strengthen the corresponding results of Kurepa and Erd\"{o}s. In this paper, we provide new and constructive proofs for these facts. In addition to practically useful recipes given here for construction of $f$, we also estimate its modulus of continuity.

math.CA↗
arXiv2024

Universal approximation theorem for neural networks with inputs from a topological vector space

We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more. We prove a universal approximation theorem for TVS-FNNs, which demonstrates their capacity to approximate any continuous function defined on this expanded input space.

cs.LG↗
arXiv2024

Addressing common misinterpretations of KART and UAT in neural network literature

This note addresses the Kolmogorov-Arnold Representation Theorem (KART) and the Universal Approximation Theorem (UAT), focusing on their frequent misinterpretations found in the neural network literature. Our remarks aim to support a more accurate understanding of KART and UAT among neural network specialists. In addition, we explore the minimal number of neurons required for universal approximation, showing that the same number of neurons needed for exact representation of functions in KART-based networks also suffices for standard multilayer perceptrons in the context of approximation.

cs.LG↗
arXiv2024

A note on the problem of straight-line interpolation by ridge functions

In this paper we discuss the problem of interpolation on straight lines by linear combinations of ridge functions with fixed directions. By using some geometry and/or systems of linear equations, we constructively prove that it is impossible to interpolate arbitrary data on any three or more straight lines by sums of ridge functions with two fixed directions. The general case with more straight lines and more directions is reduced to the problem of existence of certain sets in the union of these lines.

math.CA↗
arXiv2024

A note on the approximation by a sum of two algebras

We consider the problem of approximation of a continuous function $f$ defined on a compact metric space $X$ by elements from a sum of two algebras. We prove a de la Vall\'{e}e Poussin type theorem, which estimates the approximation error $E(f)$ from below. We also obtain a duality formula for the precise computation of $E(f)$.

math.FA↗
arXiv2023

A Chebyshev type alternation theorem for best approximation by a sum of two algebras

Let $X$ be a compact metric space, $C(X)$ be the space of continuous real-valued functions on $X$, and $A_1$, $A_2$ be two closed subalgebras of $C(X)$ containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_1+A_2$. We prove a Chebyshev type alternation theorem for a function $u_0\in A_1+A_2$ to be a best approximation to $f$.

math.FA↗
arXiv2023

On the Kolmogorov neural networks

In this paper, we show that the Kolmogorov two hidden layer neural network model with a continuous, discontinuous bounded or unbounded activation function in the second hidden layer can precisely represent continuous, discontinuous bounded and all unbounded multivariate functions, respectively.

cs.NE↗
arXiv2023

Measure theoretic results for approximation by neural networks with limited weights

In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure theoretic condition for density of such networks in the space of continuous functions. Further, we prove a density result for neural networks with a specifically constructed activation function and a fixed number of neurons.

cs.LG↗
arXiv2022

A three layer neural network can represent any multivariate function

In 1987, Hecht-Nielsen showed that any continuous multivariate function can be implemented by a certain type three-layer neural network. This result was very much discussed in neural network literature. In this paper we prove that not only continuous functions but also all discontinuous functions can be implemented by such neural networks.

cs.LG↗
arXiv2020

Notes on ridge functions and neural networks

These notes are about ridge functions. Recent years have witnessed a flurry of interest in these functions. Ridge functions appear in various fields and under various guises. They appear in fields as diverse as partial differential equations (where they are called plane waves), computerized tomography and statistics. These functions are also the underpinnings of many central models in neural networks. We are interested in ridge functions from the point of view of approximation theory. The basic goal in approximation theory is to approximate complicated objects by simpler objects. Among many classes of multivariate functions, linear combinations of ridge functions are a class of simpler functions. These notes study some problems of approximation of multivariate functions by linear combinations of ridge functions. We present here various properties of these functions. The questions we ask are as follows. When can a multivariate function be expressed as a linear combination of ridge functions from a certain class? When do such linear combinations represent each multivariate function? If a precise representation is not possible, can one approximate arbitrarily well? If well approximation fails, how can one compute/estimate the error of approximation, know that a best approximation exists? How can one characterize and construct best approximations? If a smooth function is a sum of arbitrarily behaved ridge functions, can it be expressed as a sum of smooth ridge functions? We also study properties of generalized ridge functions, which are very much related to linear superpositions and Kolmogorov's famous superposition theorem. These notes end with a few applications of ridge functions to the problem of approximation by single and two hidden layer neural networks with a restricted set of weights. We hope that these notes will be useful and interesting to both researchers and students.

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