Related papers: Differentiation via Logarithmic Expansions
Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a…
Logic-based Benders decomposition (LBBD) is a substantial generalization of classical Benders decomposition that, in principle, allows the subproblem to be any optimization problem rather than specifically a linear or nonlinear programming…
We introduce a novel distributed derivative-free optimization framework that is resilient to stragglers. The proposed method employs coded search directions at which the objective function is evaluated, and a decoding step to find the next…
Existed pre-trained models have achieved state-of-the-art performance on various text classification tasks. These models have proven to be useful in learning universal language representations. However, the semantic discrepancy between…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through…
In this paper we present a new perspective on error analysis of Legendre approximations for differentiable functions. We start by introducing a sequence of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties, such as…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
In this work, we investigate the margin-maximization bias exhibited by gradient-based algorithms in classifying linearly separable data. We present an in-depth analysis of the specific properties of the velocity field associated with…
Since its inception, Benders Decomposition (BD) has been successfully applied to a wide range of large-scale mixed-integer (linear) problems. The key element of BD is the derivation of Benders cuts, which are often not unique. In this…
This paper introduces a new approach for the computation of electromagnetic field derivatives, up to any order, with respect to the material and geometric parameters of a given geometry, in a single Finite-Difference Time-Domain (FDTD)…
In this article we show the rough outline of a computer algorithm to generate lower bounds on the exponential function of (in principle) arbitrary precision. We implemented this to generate all necessary analytic terms for the Boltzmann…
Recent technological advancements have led to a large number of patents in a diverse range of domains, making it challenging for human experts to analyze and manage. State-of-the-art methods for multi-label patent classification rely on…
Bayesian Neural Networks (BNNs) are trained to optimize an entire distribution over their weights instead of a single set, having significant advantages in terms of, e.g., interpretability, multi-task learning, and calibration. Because of…
Guessing random additive noise decoding (GRAND) is a recently proposed decoding paradigm particularly suitable for codes with short length and high rate. Among its variants, ordered reliability bits GRAND (ORBGRAND) exploits soft…
We present a dimension-incremental algorithm for the nonlinear approximation of high-dimensional functions in an arbitrary bounded orthonormal product basis. Our goal is to detect a suitable truncation of the basis expansion of the…
A generalization of the Gr\"{u}nwald difference approximation for fractional derivatives in terms of a real sequence and its generating function is presented. Properties of the generating function are derived for consistency and order of…
The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is…
We present a framework for discriminative sequence classification where the learner works directly in the high dimensional predictor space of all subsequences in the training set. This is possible by employing a new coordinate-descent…