Related papers: Beyond Taylor: Divergence-Based Functional Expansi…

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in…

This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…

In the paper, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the…

Feature Transformation (FT) crafts new features from original ones via mathematical operations to enhance dataset expressiveness for downstream models. However, existing FT methods exhibit critical limitations: discrete search struggles…

We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference…

In this paper I give an evaluation of a functional integral by means of a series in functional derivatives, first of all we propose a differential equation of first order and solve it by iterative methods, to obtain a series for the…

Density functional theory (DFT) is an indispensable ab initio method in both quantum chemistry and condensed matter physics. Based on recent advancements in reduced density matrix functional theory (RDMFT), a variant of DFT that is believed…

We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for…

We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…

We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the…

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular…

The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…

The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way.…

In this paper, we propose a numerical method of Fourier transform based on hyperfunction theory. In the proposed method, we compute analytic functions called the defining functions, which give the desired Fourier transform as a…

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. Often we have hypergeometric functions with indices linear dependent on a small parameter with respect to which…

We extend the recently introduced setting of coherent differentiation for taking into account not only differentiation, but also Taylor expansion in categories which are not necessarily (left)additive. The main idea consists in extending…

We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are…

Recent results in the theory and application of Newton-Puiseux expansions, i.e. fractional power series solutions of equations, suggest further developments within a more abstract algebraic-geometric framework, involving in particular the…

The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a…

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered…