Related papers: Infinite-dimensional generalization of Kolmogorov …
We obtain the exact values of some important approximative quantities (such as, the best approximation, the basis width, Kolmogorov's width and the best $n$-term approximation) of certain sets of images of the diagonal operators in the…
Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain…
Robust statistical inference often faces a severe computational-statistical gap when dealing with complex parameter spaces. We investigate minimax signal detection in the Gaussian sequence model under strong $\epsilon$-contamination, where…
There is a longstanding debate whether the Kolmogorov-Arnold representation theorem can explain the use of more than one hidden layer in neural networks. The Kolmogorov-Arnold representation decomposes a multivariate function into an…
This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of…
We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter…
We compare the Kolmogorov and entropy numbers of compact operators mapping from a Hilbert space into a Banach space. We then apply these general findings to embeddings between reproducing kernel Hilbert spaces and $L_\infty(\mu)$. Here we…
A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some…
The Kolmogorov-Arnold representation is a proven adequate replacement of a continuous multivariate function by an hierarchical structure of multiple functions of one variable. The proven existence of such representation inspired many…
While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This…
Kolmogorov-Arnold Network (KAN) is a network structure recently proposed by Liu et al. (2024) that offers improved interpretability and a more parsimonious design in many science-oriented tasks compared to multi-layer perceptrons. This work…
The curse of dimensionality poses a significant challenge to modern multilayer perceptron-based architectures, often causing performance stagnation and scalability issues. Addressing this limitation typically requires vast amounts of data.…
Solutions of numerous equations of mathematical physics such as elliptic, weakly singular, singular, hypersingular integral equations belong to functional classes $\bar Q^u_{r \gamma}(\Omega,1)$ and $Q^u_{r \gamma}(\Omega,1)$ defined over…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
We introduce a notion of coarse embedding at infinity into Hilbert space for metric spaces, which is a weakening of the notion of fibred coarse embedding and a far generalization of Gromov's concept of coarse embedding. It turns out that a…
In this research article, we formulate and prove multidimensional Widder--Arendt theorem and integrated form of multidimensional Widder--Arendt theorem for functions with values in sequentially complete locally convex spaces. Established…
The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical…
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from…
This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity…
This monograph is devoted to the theory of entire functions of several variables. A definition of bounded index was supposed by B. Lepson. We generalised his definition for several variables and obtained criteria of L-index boundedness in…