Related papers: Bochner integrals and neural networks

Hille's theorem is a powerful classical result in vector measure theory. It asserts that the application of a closed, unbounded linear operator commutes with strong/Bochner integration of functions taking values in a Banach space. This note…

Deep learning is computationally intensive, with significant efforts focused on reducing arithmetic complexity, particularly regarding energy consumption dominated by data movement. While existing literature emphasizes inference, training…

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…

Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address…

We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.

Neural networks (NNs) whose subnetworks implement reusable functions are expected to offer numerous advantages, including compositionality through efficient recombination of functional building blocks, interpretability, preventing…

We introduce new weighted $L^p$-type spaces defined in terms of weight function matrices and characterize the inclusion relations in terms of the defining matrices. Moreover, we provide a detailed study concerning the coincidence with the…

The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.

We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the…

A nonlinear equation in a Banach space is written as a linear equation with a linear operator depending on the unknown solution. This method, which we call a global linearization method, differs essentially from the local linearization…

We define and establish the conditions for `equivalent neural networks' - neural networks with different weights, biases, and threshold functions that result in the same associated function. We prove that given a neural network…

The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint…

We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian…

We show that some previous results concerning the boundedness of differentiation and integration operators on weighted spaces given by radial weights in the unit disk or the complex plane might fail without some natural additional…

A neural network computes a function. A central property of neural networks is that they are "universal approximators:" for a given continuous function, there exists a neural network that can approximate it arbitrarily well, given enough…

The objective of this paper is to introduce and study a complicated nonlinear system, called coupled variational-hemivariational inequalities, which is described by a highly nonlinear coupled system of inequalities on Banach spaces. We…

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…

This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic…

This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…

In many computational problems in engineering and science, function or model differentiation is essential, but also integration is needed. An important class of computational problems include so-called integro-differential equations which…