Neural network approximation of continuous functions in high dimensions with applications to inverse problems

Abstract

The remarkable successes of neural networks in a huge variety of inverse problems has fueled their adoption in disciplines ranging from medical imaging to seismic analysis over the past decade. However, the high dimensionality of such inverse problems has simultaneously left current theory, which predicts that networks should scale exponentially in the dimension of the problem, unable to explain why the seemingly small networks used in these settings work as well as they do in practice. To reduce this gap between theory and practice, a general method for bounding the complexity required for a neural network to approximate a Hölder continuous function (as well as uniformly continuous functions that admit a specified modulus of continuity) defined on a high-dimensional set with a low-complexity structure is provided herein. The idea for the approach is based on the observation that the existence of a linear Johnson–Lindenstrauss embedding A∈Rd×D of a given high-dimensional set S⊂RD into a low dimensional cube [−M,M]d implies that for any Hölder (or uniformly) continuous function f:S→Rp, there exists a Hölder (or uniformly) continuous function g:[−M,M]d→Rp such that g(Ax)=f(x) for all x∈S. Hence, if one has a neural network which approximates g:[−M,M]d→Rp, then a layer can be added that implements the JL embedding A to obtain a neural network that approximates f:S→Rp. By pairing JL embedding results along with results on approximation of Hölder (or uniformly) continuous functions by neural networks, one then obtains results which bound the complexity required for a neural network to approximate Hölder (or uniformly) continuous functions on high dimensional sets. The end result is a general theoretical framework which can then be used to better explain the observed empirical successes of smaller networks in a wider variety of inverse problems than current theory allows.