Related papers: Automatic Differentiation With Higher Infinitesima…
Modelers use automatic differentiation (AD) of computation graphs to implement complex Deep Learning models without defining gradient computations. Stochastic AD extends AD to stochastic computation graphs with sampling steps, which arise…
Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators…
Determining physical properties inside an object without access to direct measurements of target regions can be formulated as a specific type of \textit{inverse problem}. One of such problems is applied in \textit{Electrical Impedance…
We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of…
Formal transformations somehow resembling the usual derivative are surprisingly common in computer science, with two notable examples being derivatives of regular expressions and derivatives of types. A newcomer to this list is the…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
A higher-order accurate finite element method is proposed which uses automatically generated meshes based on implicit level-set data for the description of boundaries and interfaces in two and three dimensions. The method is an alternative…
In this paper, we introduce the concept of a "von Neumann regular $\mathcal{C}^{\infty}$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can…
Border bases are a generalization of Gr\"obner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra.…
Automatic differentiation is involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic…
Let (W,S) be a Coxeter system of affine type D, and let TL(W) the corresponding generalized Temperley-Lieb algebra. In this extended abstract we define an infinite dimensional associative algebra made of decorated diagrams which is…
The modified Cholesky decomposition (MCD) is an efficient technique for estimating a covariance matrix. However, it is known that the MCD technique often requires a pre-specified variable ordering in the estimation procedure. In this work,…
Differential machine learning combines automatic adjoint differentiation (AAD) with modern machine learning (ML) in the context of risk management of financial Derivatives. We introduce novel algorithms for training fast, accurate pricing…
The assumption of conditional independence among observed variables, primarily used in the Variational Autoencoder (VAE) decoder modeling, has limitations when dealing with high-dimensional datasets or complex correlation structures among…
The explication of Convolutional Neural Networks (CNN) through xAI techniques often poses challenges in interpretation. The inherent complexity of input features, notably pixels extracted from images, engenders complex correlations.…
We define the algebraic Dirac induction map $\Ind_D$ for graded affine Hecke algebras. The map $\Ind_D$ is a Hecke algebra analog of the explicit realization of the Baum-Connes assembly map in the $K$-theory of the reduced $C^*$-algebra of…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to $\mathcal{O}(d^{k})$ scaling of the derivative tensor size and the…
The method of direct calculation of the group of $\mathbb R$-algebra automorphisms of a Weil algebra is presented in detail. The paper is focused on the case of a one-componental group and presents two cases of values of the determinant of…
We provide a computable criterion for selecting among Fourier, wavelet, and time-frequency analysis by extending the algebraic diversity (AD) framework to Lie groups acting on $L^2(\mathbb{R})$. To our knowledge, there is no other criterion…