Related papers: Jacobians and Gradients for Cartesian Differential…
Deep neural networks are notorious for defying theoretical treatment. However, when the number of parameters in each layer tends to infinity, the network function is a Gaussian process (GP) and quantitatively predictive description is…
We construct a 2-category of differential graded schemes. The local affine models in this theory are differential graded algebras, which are graded commutative with unit over a field of characteristic zero, are concentrated in non-positive…
We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of…
Building on Simpson's original definition over the complex numbers, we introduce the notion of restricted sheaf $\Lambda$ of rings of differential operators on a variety defined over a field of positive characteristic. We define the notion…
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and…
Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y).…
In this paper we propose two proximal gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either…
This article presents a unified approach to simultaneously compute the Jacobians of several singular matrix transformations in the real, complex, quaternion and octonion cases. Formally, these Jacobians are obtained for real normed division…
In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a…
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial…
Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and…
Derivatives of computer graphics, image processing, and deep learning algorithms have tremendous use in guiding parameter space searches, or solving inverse problems. As the algorithms become more sophisticated, we no longer only need to…
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the…
Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided…
Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the…
In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $TT^*Q$…
A Deep Neural Network (DNN) is a composite function of vector-valued functions, and in order to train a DNN, it is necessary to calculate the gradient of the loss function with respect to all parameters. This calculation can be a…
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…
By using the Hadamard matrix product concept, this paper introduces two generalized matrix formulation forms of numerical analogue of nonlinear differential operators. The SJT matrix-vector product approach is found to be a simple,…
Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum…