Related papers: Jacobians and Gradients for Cartesian Differential…
Computing the Jacobian of the solution of an optimization problem is a central problem in machine learning, with applications in hyperparameter optimization, meta-learning, optimization as a layer, and dataset distillation, to name a few.…
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…
In computational pathology, cancer grading has been mainly studied as a categorical classification problem, which does not utilize the ordering nature of cancer grades such as the higher the grade is, the worse the cancer is. To incorporate…
We establish sharp trace- and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov…
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…
Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us…
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…
Jacobi-type algorithms for simultaneous approximate diagonalization of real (or complex) symmetric tensors have been widely used in independent component analysis (ICA) because of their good performance. One natural way of choosing the…
Recursive coalgebras provide an elegant categorical tool for modelling recursive algorithms and analysing their termination and correctness. By considering coalgebras over categories of suitably indexed families, the correctness of the…
Fractional-order differentiation has many characteristics different from integer-order differentiation. These characteristics can be applied to the optimization algorithms of artificial neural networks to obtain better results. However, due…
Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
Derivations provide a way of transporting ideas from the calculus of manifolds to algebraic settings where there is no sensible notion of limit. In this paper, we consider derivations in certain monoidal categories, called codifferential…
We outline a general algorithm for computing an explicit model over a number field of any curve of genus 2 whose (unpolarized) Jacobian is isomorphic to the product of two elliptic curves with CM by the same order in an imaginary quadratic…
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…
Large language models (LLMs) make next-token predictions based on clues present in their context, such as semantic descriptions and in-context examples. Yet, elucidating which prior tokens most strongly influence a given prediction remains…
Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic…
This is the fourth (and last) prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to…
Deep learning has been widely used for supervised learning and classification/regression problems. Recently, a novel area of research has applied this paradigm to unsupervised tasks; indeed, a gradient-based approach extracts, efficiently…