Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?

We study the problem of approximating compactly-supported integrable functions while implementing their support set using feedforward neural networks. Our first main result transcribes this "structured" approximation problem into a universality problem. We do this by constructing a refinement of the usual topology on the space $L^1_{\operatorname{loc}}(\mathbb{R}^d,\mathbb{R}^D)$ of locally-int…