Related papers: New universal operator approximation theorem for e…

We introduce a new and extensive theory of noncommutative convexity along with a corresponding theory of noncommutative functions. We establish noncommutative analogues of the fundamental results from classical convexity theory, and apply…

We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are illustrated with common examples of the theory of umbral…

Averaged operators have played an important role in fixed point theory in Hilbert spaces. They emerged as a necessity to obtain solutions to fixed point problems where the underlying operator is not contractive and thus renders Banach fixed…

Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational…

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable…

In the present article, we introduce and study the behaviour of the new family of exponential type neural network operators activated by the sigmoidal functions. We establish the point-wise and uniform approximation theorems for these NN…

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have…

We study feedforward neural networks with inputs from a topological space (TFNNs). We prove a universal approximation theorem for shallow TFNNs, which demonstrates their capacity to approximate any continuous function defined on this…

In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by sigmoidal function. Other than a uniform convergence theorem…

This work investigates the properties of the proximity operator for quasar-convex functions and establishes the convergence of the proximal point algorithm to a global minimizer with a particular focus on its convergence rate. In…

Deep neural networks are powerful learning models that achieve state-of-the-art performance on many computer vision, speech, and language processing tasks. In this paper, we study a fundamental question that arises when designing deep…

In established network architectures, shortcut connections are often used to take the outputs of earlier layers as additional inputs to later layers. Despite the extraordinary effectiveness of shortcuts, there remain open questions on the…

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically the cascade algorithm in wavelet theory. Let $ H $ be a Hilbert space, and let $ \pi $ be a representation…

There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition $\OO$ of the universal function $f$. In another kind the universal…

Approximate learning machines have become popular in the era of small devices, including quantised, factorised, hashed, or otherwise compressed predictors, and the quest to explain and guarantee good generalisation abilities for such…

Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential…

For Banach spaces $X,Y,$ we consider a distance problem in the space of bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper \cite{RAO21}, we obtain sufficient conditions so that for a compact operator…

In this paper, we propose a DeepONet structure with causality to represent the causal linear operators between Banach spaces of time-dependent signals. The theorem of universal approximations to nonlinear operators proposed in…

We introduce \emph{in-context operator learning on probability measure spaces} for optimal transport (OT). The goal is to learn a single solution operator that maps a pair of distributions to the OT map, using only few-shot samples from…

The convolution operator is the fundamental building block of modern convolutional neural networks (CNNs), owing to its simplicity, translational equivariance, and efficient implementation. However, its structure as a fixed, linear,…