  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-integrable
functions in which compactly-supported functions can only be approximated in
$L^1$-norm by functions with matching discretized support. We establish the
universality of ReLU feedforward networks with bilinear pooling layers in this
refined topology. Consequentially, we find that ReLU feedforward networks with
bilinear pooling can approximate compactly supported functions while
implementing their discretized support. We derive a quantitative uniform
version of our universal approximation theorem on the dense subclass of
compactly-supported Lipschitz functions. This quantitative result expresses the
depth, width, and the number of bilinear pooling layers required to construct
this ReLU network via the target function's regularity, the metric capacity and
diameter of its essential support, and the dimensions of the inputs and output
spaces. Conversely, we show that polynomial regressors and analytic feedforward
networks are not universal in this space.
