Bochner integrals and neural networks
A Bochner integral formula is derived that represents a function in terms of weights and a parametrized family of functions. Comparison is made to pointwise formulations, norm inequalities relating pointwise and Bochner integrals are established, variation-spaces and tensor products are studied, and examples are presented. The paper develops a functional analytic theory of neural networks and shows that variation spaces are Banach spaces.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Shrinkage Estimation of Higher Order Bochner Integrals
We consider shrinkage estimation of higher order Hilbert space valued Bochner integrals in a non-parametric setting. We propose estimators that shrink the $U$-statistic estimator of the Bochner integral towards a pre-spe…
Approximations with deep neural networks in Sobolev time-space
Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables.…
On Bochner's and Polya's Characterizations of Positive-Definite Kernels and the Respective Random Feature Maps
Positive-definite kernel functions are fundamental elements of kernel methods and Gaussian processes. A well-known construction of such functions comes from Bochner's characterization, which connects a positive-definite …
Gaussian ProcessesSpace-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces
Approximation capabilities of shallow neural networks (SNNs) form an integral part in understanding the properties of deep neural networks (DNNs). In the study of these approximation capabilities some very popular classe…
Efficient AI-Inspired Reduction of Feynman Integrals via Tube Seeding
In this paper, we use machine learning to discover a new seeding strategy for integration-by-parts reduction of Feynman integrals, which is a frequent bottleneck in state-of-the-art calculations in theoretical particle a…