Related papers: Combining fixpoint and differentiation theory
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
Logic programming with fixed-point definitions is a useful extension of traditional logic programming. Fixed-point definitions can capture simple model checking problems and closed-world assumptions. Its operational semantics is typically…
Fractional gradient descent has been studied extensively, with a focus on its ability to extend traditional gradient descent methods by incorporating fractional-order derivatives. This approach allows for more flexibility in navigating…
A class of improved estimators is proposed for N-point correlation functions of galaxy clustering, and for discrete spatial random processes in general. In the limit of weak clustering, the variance of the unbiased estimator converges to…
We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars,…
We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
We consider the operation of intersecting with a locally principal Cartier divisor (i.e., a Cartier divisor which is principal on some neighborhood of its support). We describe this operation explicitly on the level of cycles and rational…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in…
This paper is concerned with the approximation of solutions to a class of second order non linear abstract differential equations. The finite-dimensional approximate solutions of the given system are built with the aid of the projection…
Whenever eye movements are measured, a central part of the analysis has to do with where subjects fixate, and why they fixated where they fixated. To a first approximation, a set of fixations can be viewed as a set of points in space: this…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
Cartesian reverse derivative categories (CRDCs) provide an axiomatic generalization of the reverse derivative, which allows generalized analogues of classic optimization algorithms such as gradient descent to be applied to a broad class of…
This thesis develops the translation between category theory and computational linguistics as a foundation for natural language processing. The three chapters deal with syntax, semantics and pragmatics. First, string diagrams provide a…
We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname{DCF}_p$ and…
Categorization axioms have been proposed to axiomatizing clustering results, which offers a hint of bridging the difference between human recognition system and machine learning through an intuitive observation: an object should be assigned…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…