Related papers: Approximation with Neural Networks in Variable Leb…

In this paper, we first give a sufficiently condition for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by translation operator. Then we obtain a criterion for precompactness in the matrix-weighted Lebesgue…

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e…

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…

In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…

The capability of recurrent neural networks to approximate trajectories of a random dynamical system, with random inputs, on non-compact domains, and over an indefinite or infinite time horizon is considered. The main result states that…

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both…

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input…

Parametric optimization solves a family of optimization problems as a function of parameters. It is a critical component in situations where optimal decision making is repeatedly performed for updated parameter values, but computation…

Our aim is to characterize the Lipschitz functions by variable exponent Lebesgue spaces. We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy-Littlewood maximal function and sharp maximal…

Despite the fact that generative models are extremely successful in practice, the theory underlying this phenomenon is only starting to catch up with practice. In this work we address the question of the universality of generative models:…

This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for…

Permutation-invariant, -equivariant, and -covariant functions and anti-symmetric functions are important in quantum physics, computer vision, and other disciplines. Applications often require most or all of the following properties: (a) a…

In comparison to classical shallow representation learning techniques, deep neural networks have achieved superior performance in nearly every application benchmark. But despite their clear empirical advantages, it is still not well…

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with…

We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed…

The problem of extending a function $f$ defined on a training data $\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For $\mathbb{X}$ embedded…

Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions.…

Neural nets are known to be universal approximators. In particular, formal neurons implementing wavelets have been shown to build nets able to approximate any multidimensional task. Such very specialized formal neurons may be, however,…

For the past 30 years or so, machine learning has stimulated a great deal of research in the study of approximation capabilities (expressive power) of a multitude of processes, such as approximation by shallow or deep neural networks,…

One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…