  We investigate to what extent it is possible to solve linear inverse problems
with $ReLu$ networks. Due to the scaling invariance arising from the linearity,
an optimal reconstruction function $f$ for such a problem is positive
homogeneous, i.e., satisfies $f(\lambda x) = \lambda f(x)$ for all non-negative
$\lambda$. In a $ReLu$ network, this condition translates to considering
networks without bias terms. We first consider recovery of sparse vectors from
few linear measurements. We prove that $ReLu$- networks with only one hidden
layer cannot even recover $1$-sparse vectors, not even approximately, and
regardless of the width of the network. However, with two hidden layers,
approximate recovery with arbitrary precision and arbitrary sparsity level $s$
is possible in a stable way. We then extend our results to a wider class of
recovery problems including low-rank matrix recovery and phase retrieval.
Furthermore, we also consider the approximation of general positive homogeneous
functions with neural networks. Extending previous work, we establish new
results explaining under which conditions such functions can be approximated
with neural networks. Our results also shed some light on the seeming
contradiction between previous works showing that neural networks for inverse
problems typically have very large Lipschitz constants, but still perform very
well also for adversarial noise. Namely, the error bounds in our expressivity
results include a combination of a small constant term and a term that is
linear in the noise level, indicating that robustness issues may occur only for
very small noise levels.
