Neural Network Approximation of Lipschitz Functions in High Dimensions with Applications to Inverse Problems

The remarkable successes of neural networks in a huge variety of inverse problems have 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 Lipschitz function on a high-dimensional set with a low-complexity structure is provided herein. The approach is based on the observation that the existence of a linear Johnson-Lindenstrauss embedding $\mathbf{A} \in \mathbb{R}^{d \times D}$ of a given high-dimensional set $\mathcal{S} \subset \mathbb{R}^D$ into a low dimensional cube $[-M,M]^d$ implies that for any Lipschitz function $f : \mathcal{S}\to \mathbb{R}^p$, there exists a Lipschitz function $g : [-M,M]^d \to \mathbb{R}^p$ such that $g(\mathbf{A}\mathbf{x}) = f(\mathbf{x})$ for all $\mathbf{x} \in \mathcal{S}$. Hence, if one has a neural network which approximates $g : [-M,M]^d \to \mathbb{R}^p$, then a layer can be added which implements the JL embedding $\mathbf{A}$ to obtain a neural network which approximates $f : \mathcal{S} \to \mathbb{R}^p$. By pairing JL embedding results along with results on approximation of Lipschitz functions by neural networks, one then obtains results which bound the complexity required for a neural network to approximate Lipschitz 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.