Related papers: A Language for Evaluating Derivatives of Functiona…
We give a simple, direct and reusable logical relations technique for languages with term and type recursion and partially defined differentiable functions. We demonstrate it by working out the case of Automatic Differentiation (AD)…
We present a probabilistic version of PCF, a well-known simply typed universal functional language. The type hierarchy is based on a single ground type of natural numbers. Even if the language is globally call-by-name, we allow a…
This paper introduces a new derivative parsing algorithm for recognition of parsing expression grammars. Derivative parsing is shown to have a polynomial worst-case time bound, an improvement on the exponential bound of the recursive…
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…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
Solving partial differential equations (PDEs) efficiently is essential for analyzing complex physical systems. Recent advancements in leveraging deep learning for solving PDE have shown significant promise. However, machine learning…
In functional data analysis, replicate observations of a smooth functional process and its derivatives offer a unique opportunity to flexibly estimate continuous-time ordinary differential equation models. Ramsay (1996) first proposed to…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
This paper enriches preexisting satisfiability tests for unquantified languages, which in turn augment a fragment of Tarski's elementary algebra with unary real functions possessing a continuous first derivative. Two sorts of individual…
Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and…
Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the…
Mapping the functional connectome has the potential to uncover key insights into brain organisation. However, existing workflows for functional connectomics are limited in their adaptability to new data, and principled workflow design is a…
We introduce a new setting, the category of $\omega$PAP spaces, for reasoning denotationally about expressive differentiable and probabilistic programming languages. Our semantics is general enough to assign meanings to most practical…
Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently,…
We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher-order language with algebraic data types and we characterise it as the unique structure-preserving macro given a…
We face the problems of correctness, optimality and precision for the static analysis of logic programs, using the theory of abstract interpretation. We propose a framework with a denotational, goal-dependent semantics equipped with two…
The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes…
This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional…
We show how to define forward- and reverse-mode automatic differentiation source-code transformations or on a standard higher-order functional language. The transformations generate purely functional code, and they are principled in the…