Related papers: Approximation by Neural Network Sampling Operators…
We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to…
The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by…
Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the…
In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of…
Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in…
The paper contains approximation guarantees for neural networks that are trained with gradient flow, with error measured in the continuous $L_2(\mathbb{S}^{d-1})$-norm on the $d$-dimensional unit sphere and targets that are Sobolev smooth.…
This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special…
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular…
We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in…
The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…
In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions…
Neural networks have shown great predictive power when dealing with various unstructured data such as images and natural languages. The Bayesian neural network captures the uncertainty of prediction by putting a prior distribution for the…
Comparing the internal representations of neural networks is a central goal in both neuroscience and machine learning. Standard alignment metrics operate on raw neural activations, implicitly assuming that similar representations produce…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
This paper derives rates of convergence of certain approximations of the Koopman operators that are associated with discrete, deterministic, continuous semiflows on a complete metric space $(X,d_X)$. Approximations are constructed in terms…
We study approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ As an application of the results obtained, we characterize smoothness of a bounded linear…
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
Complex-valued neural networks (CVNNs) have recently shown promising empirical success, for instance for increasing the stability of recurrent neural networks and for improving the performance in tasks with complex-valued inputs, such as in…