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We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a measure sup
Externí odkaz:
http://arxiv.org/abs/2410.09214
Autor:
Ismailov, Vugar
This note addresses the Kolmogorov-Arnold Representation Theorem (KART) and the Universal Approximation Theorem (UAT), focusing on their common misinterpretations in some papers related to neural network approximation. Our remarks aim to support a mo
Externí odkaz:
http://arxiv.org/abs/2408.16389
We highlight a perhaps important but hitherto unobserved insight: The attention module in a transformer is a smoothed cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old
Externí odkaz:
http://arxiv.org/abs/2408.09624
Autor:
Akhmedov, Azer, Ismailov, Vugar
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 inter
Externí odkaz:
http://arxiv.org/abs/2408.06443
Recently Tomasz Natkaniec in [On lineability of families of non-measurable functions of two variable. Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 33, 10, 2021] studied the lineability problem for several classes of n
Externí odkaz:
http://arxiv.org/abs/2309.00830
Autor:
van Nuland, Teun D. H.
The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all
Externí odkaz:
http://arxiv.org/abs/2308.03812
To present Mercer large-scale kernel machines from a ridge function perspective, we recall the results by Lin and Pinkus from {\it Fundamentality of ridge functions}. We consider the main result of the recent paper by Rachimi and Recht, 2008, {\it Ra
Externí odkaz:
http://arxiv.org/abs/2307.11925
Autor:
Poluektov, Michael, Polar, Andrew
It is known that any continuous multivariate function can be represented exactly by a composition functions of a single variable - the so-called Kolmogorov-Arnold representation. It can be a convenient tool for tasks where it is required to obtain a
Externí odkaz:
http://arxiv.org/abs/2305.08194
Autor:
Chebotarev, Pavel
We consider the problem of extending a function $f^{}_P$ defined on a subset $P$ of an arbitrary set $X$ to $X$ strictly monotonically with respect to a preorder $\succcurlyeq$ defined on $X$, without imposing continuity constraints. We show that whe
Externí odkaz:
http://arxiv.org/abs/2212.03394
Autor:
Gorokhovik, Valentin V.
The goal of the paper is to study the particular class of regularly ${\mathcal{H}}$-convex functions, when ${\mathcal{H}}$ is the set ${\mathcal{L}\widehat{C}}(X,{\mathbb{R}})$ of real-valued Lipschitz continuous classically concave functions defined
Externí odkaz:
http://arxiv.org/abs/2208.01541