Related papers: Universal Pad\'e approximants on simply connected …
We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are…
Let $\omega_0,\dots,\omega_M$ be complex numbers. If $H_0,\dots,H_M$ are polynomials of degree at most $\rho_0,\dots,\rho_M$, and $G(z)=\sum_{m=0} ^M H_m(z) (1-z)^{\omega_m}$ has a zero at $z=0$ of maximal order (for the given…
We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra.…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
We establish fundamental mathematical limits on universal approximation theorem (UAT) system alignment by proving that catastrophic failures are an inescapable feature of any useful computational system. Our central thesis is that for any…
Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…
We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more…
Cartesian differential categories provide a categorical framework for multivariable differential calculus and also the categorical semantics of the differential $\lambda$-calculus. Taylor series expansion is an important concept for both…
This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is…
The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in…
We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…
The main purpose of this paper is to compare the convergence properties of Pad\'e approximants and rational Hermite-Pad\'e approximants for some model class of multivalued analytic functions based of the inverse Zhoukovsky transform. We…
The form factor of the unitary group U(N) endowed with the Haar measure characterizes the correlations within the spectrum of a typical unitary matrix. It can be decomposed into a sum over pairs of ``periodic orbits'', where by periodic…
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 explain that, like the usual Pad\'e approximants, the barycentric Pad\'e approximants proposed recently by Brezinski and Redivo-Zaglia can diverge. More precisely, we show that for every polynomial P there exists a power series S, with…
We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\varrho:\mathbb{C}\to \mathbb{C}$ that have the property…
Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their…
We introduce a class of fully-connected neural networks whose activation functions, rather than being pointwise, rescale feature vectors by a function depending only on their norm. We call such networks radial neural networks, extending…
Let $(\tau_n)_n$ be a sequence of real numbers in $(1,+\infty)$. Using potential theoretic methods, we prove quantitative results - Bernstein-Walsh type theorems - about uniform approximation by polynomials of the form $\sum_{k=\lfloor…
On a compact subset of the plane with connected complement, is it possible to uniformly approximate a continuous function, holomorphic and non-vanishing on the interior, with polynomials non-vanishing on the entire compact set? In this…