Related papers: How to Define Automatic Differentiation
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
The goal of this thesis is threefold: first, to provide a general semantic setting for reasoning about incremental computation. Second, to establish and clarify the connection between derivatives in the incremental sense and derivatives in…
Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of…
There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first…
The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…
This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…
Kjolstad et. al. proposed a tensor algebra compiler. It takes expressions that define a tensor element-wise, such as $f_{ij}(a,b,c,d) = \exp\left[-\sum_{k=0}^4 \left((a_{ik}+b_{jk})^2\, c_{ii} + d_{i+k}^3 \right) \right]$, and generates the…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
We present a technique designed to automatically compute predicate abstractions for dense real-timed models represented as networks of timed automata. We use the CIPM algorithm in our previous work which computes new invariants for timed…
This paper presents a novel approach to automatically solving arithmetic word problems. This is the first algorithmic approach that can handle arithmetic problems with multiple steps and operations, without depending on additional…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. We discuss how to estimate errors of a general function…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are…
We develop nested automatic differentiation (AD) algorithms for exact inference and learning in integer latent variable models. Recently, Winner, Sujono, and Sheldon showed how to reduce marginalization in a class of integer latent variable…
We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can…
This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…
We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.
In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action…