Sparse recovery from saturated measurements

A novel theory of sparse recovery is presented in order to bridge the standard compressive sensing framework and the one-bit compressive sensing framework. In the former setting, sparse vectors observed via few linear measurements can be reconstructed exactly. In the latter setting, the linear measurements are only available through their signs, so exact reconstruction of sparse vectors is replaced by estimation of their directions. In the hybrid setting introduced here, a linear measurement is conventionally acquired if is not too large in absolute value, but otherwise it is seen as saturated to plus-or-minus a given threshold. Intuition suggests that sparse vectors of small magnitude should be exactly recoverable, since saturation would not occur, and that sparse vectors of larger magnitude should be accessible though more than just their directions. The purpose of the article is to confirm this intuition and to justify rigorously the following informal statement: measuring at random with Gaussian vectors and reconstructing via an `1-minimization scheme, it is highly likely that all sparse vectors are faithfully estimated from their saturated measurements as long as the number of saturated measurements marginally exceeds the sparsity level. Faithful estimation means exact reconstruction in a small-magnitude regime and control of the relative reconstruction error in a larger-magnitude regime.