  In compressed sensing, a small number of linear measurements can be used to
reconstruct an unknown signal. Existing approaches leverage assumptions on the
structure of these signals, such as sparsity or the availability of a
generative model. A domain-specific generative model can provide a stronger
prior and thus allow for recovery with far fewer measurements. However, unlike
sparsity-based approaches, existing methods based on generative models
guarantee exact recovery only over their support, which is typically only a
small subset of the space on which the signals are defined. We propose
Sparse-Gen, a framework that allows for sparse deviations from the support set,
thereby achieving the best of both worlds by using a domain specific prior and
allowing reconstruction over the full space of signals. Theoretically, our
framework provides a new class of signals that can be acquired using compressed
sensing, reducing classic sparse vector recovery to a special case and avoiding
the restrictive support due to a generative model prior. Empirically, we
observe consistent improvements in reconstruction accuracy over competing
approaches, especially in the more practical setting of transfer compressed
sensing where a generative model for a data-rich, source domain aids sensing on
a data-scarce, target domain.
