Lower Bounds for Approximation of Some Classes of Lebesgue Measurable Functions by Sigmoidal Neural Networks

摘要

We propose a general method for estimating the distance between a compact subspace K of the space L~1([0, 1]~s) of Lebesgue measurable functions defined on the hypercube [0, 1]~s, and the class of functions computed by artificial neural networks using a single hidden layer, each unit evaluating a sigmoidal activation function. Our lower bounds are stated in terms of an invariant that measures the oscillations of functions of the space K around the origin. As an application we estimate the minimal number of neurons required to approximate bounded functions satisfying uniform Lipschitz conditions of order α with accuracy ∈.