Lipschitz Widths

This paper introduces a measure, called Lipschitz widths, of the optimal performance possible of certain nonlinear methods of approximation. Notably, those Lipschitz widths provide a theoretical benchmark for the approximation quality achieved via deep neural networks. The paper also discusses basic properties of the Lipschitz widths and their relation to entropy numbers and other well-known widths such as the Kolmogorov and the stable manifold widths. We show that Lipschitz widths with fixed Lipschitz constant and entropy numbers decay very similar, while when the Lipschitz constant grows with n , the Lipschitz width could be much smaller than the entropy numbers.