Learning a Sparse Representation of Barron Functions with the Inverse
  Scale Space Flow
Tjeerd Jan Heeringa Tim Roith Christoph Brune Martin Burger
This paper presents a method for finding a sparse representation of Barron
functions. Specifically, given an $L^2$ function $f$, the inverse scale space
flow is used to find a sparse measure $\mu$ minimising the $L^2$ loss between
the Barron function associated to the measure $\mu$ and the function $f$. The
convergence properties of this method are analysed in an ideal setting and in
the cases of measurement noise and sampling bias. In an ideal setting the
objective decreases strictly monotone in time to a minimizer with
$\mathcal{O}(1/t)$, and in the case of measurement noise or sampling bias the
optimum is achieved up to a multiplicative or additive constant. This
convergence is preserved on discretization of the parameter space, and the
minimizers on increasingly fine discretizations converge to the optimum on the
full parameter space.