  We study compressive sensing with a deep generative network prior. Initial
theoretical guarantees for efficient recovery from compressed linear
measurements have been developed for signals in the range of a ReLU network
with Gaussian weights and logarithmic expansivity: that is when each layer is
larger than the previous one by a logarithmic factor. It was later shown that
constant expansivity is sufficient for recovery. It has remained open whether
the expansivity can be relaxed, allowing for networks with contractive layers
(as often the case of real generators). In this work we answer this question,
proving that a signal in the range of a Gaussian generative network can be
recovered from few linear measurements provided that the width of the layers is
proportional to the input layer size (up to log factors). This condition allows
the generative network to have contractive layers. Our result is based on
showing that Gaussian matrices satisfy a matrix concentration inequality which
we term Range Restricted Weight Distribution Condition (R2WDC) and that weakens
the Weight Distribution Condition (WDC) upon which previous theoretical
guarantees were based. The WDC has also been used to analyze other signal
recovery problems with generative network priors. By replacing the WDC with the
R2WDC, we are able to extend previous results for signal recovery with
expansive generative network priors to non-expansive ones. We discuss these
extensions for phase retrieval, denoising, and spiked matrix recovery.
