Related papers: Reverse Tangent Categories

To achieve near-zero training error in a classification problem, the layers of a feed-forward network have to disentangle the manifolds of data points with different labels, to facilitate the discrimination. However, excessive class…

Cartesian differential categories were introduced to provide an abstract axiomatization of categories of differentiable functions. The fundamental example is the category whose objects are Euclidean spaces and whose arrows are smooth maps.…

This paper explores a new perspective on the universality of the vertical lift in tangent categories by presenting a categorification of the dimension of smooth manifolds. The universality of the vertical lift is a key part of the axioms of…

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, $\mathscr{P}$-affine schemes can be defined as objects of the opposite of the category of algebras over an operad…

We develop foundations for oriented category theory, an extension of $(\infty,\infty)$-category theory obtained by systematic usage of the Gray tensor product, in order to study lax phenomena in higher category theory. As categorical…

It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…

Many machine learning problems involve regressing variables on a non-Euclidean manifold -- e.g. a discrete probability distribution, or the 6D pose of an object. One way to tackle these problems through gradient-based learning is to use a…

In real-world scenarios classification models are often required to perform robustly when predicting samples belonging to classes that have not appeared during its training stage. Open Set Recognition addresses this issue by devising models…

In this work we introduce a convolution operation over the tangent bundle of Riemannian manifolds exploiting the Connection Laplacian operator. We use the convolution to define tangent bundle filters and tangent bundle neural networks…

The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having…

In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which…

In this article a surprising result is demonstrated using the neural tangent kernel. This kernel is defined as the inner product of the vector of the gradient of an underlying model evaluated at training points. This kernel is used to…

Variational quantum circuits are used in quantum machine learning and variational quantum simulation tasks. Designing good variational circuits or predicting how well they perform for given learning or optimization tasks is still unclear.…

Gradient-descent methods have exhibited fast and reliable performance for image alignment in the facial domain, but have largely been ignored by the broader vision community. They require the image function be smooth and (numerically)…

Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain…

We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…

Many neural network architectures are known to be Turing Complete, and can thus, in principle implement arbitrary algorithms. However, Transformers are unique in that they can implement gradient-based learning algorithms under simple…

This paper addresses the limitations of Physics-Informed Neural Networks for time-dependent problems by introducing a tangent bundle learning framework. Instead of directly approximating the solution, we parameterize its temporal derivative…

In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual…