Lipschitz widths

TL;DR

This paper introduces Lipschitz widths as a new measure to evaluate the optimal approximation performance of nonlinear methods, relating it to existing widths and establishing it as a benchmark for deep neural network approximation quality.

Contribution

It defines Lipschitz widths and explores their relation to entropy and other widths, providing a theoretical benchmark for neural network approximation.

Findings

Lipschitz widths are related to entropy and Kolmogorov widths.

They serve as a theoretical benchmark for deep neural network approximation.

The paper establishes connections between Lipschitz widths and existing approximation measures.

Abstract

This paper introduces a measure, called Lipschitz widths, of the optimal performance possible of certain nonlinear methods of approximation. It discusses their relation to entropy numbers and other well known widths such as the Kolmogorov and the stable manifold widths. It also shows that the Lipschitz widths provide a theoretical benchmark for the approximation quality achieved via deep neural networks.