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, we provide 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. The approach is based on the fact that many sets of interest in high dimensions have low-distortion linear embeddings into lower dimensional spaces. We can exploit this fact to show that the size of a neural network needed to approximate a Lipschitz function on a low-complexity set in a high dimensional space grows exponentially with the dimension of its low-distortion embedding, not the dimension of the space it lies in.