Performance Analysis of One-bit Group-sparse Signal Reconstruction

We consider the reconstruction of group-sparse vectors from sign measurements of random projections. In particular, we establish conditions on the number of measurements under which such signal ensembles can be recovered up to a prescribed accuracy. The results rely on a mixed restricted isometry property as first employed by Foucart in the context of 1-bit compressed sensing, as well as certain results on random hyperplane tessellations. The paper fills a gap in the literature by establishing that group-sparse signals can be estimated from 1-bit observations with the same number of measurements as required for the reconstruction of block-sparse signals from unquantized measurements. We confirm the correct behavior of the recovery schemes in a series of numerical experiments.