Reconstruction of Sparse Signals From l1 Dimensionality-Reduced Cauchy Random Projections

Dimension reduction methods via linear random projections are used in numerous applications including data mining, information retrieval and compressive sensing (CS). While CS has traditionally relied on normal random projections, corresponding to l(2) distance preservation, a large body of work has emerged for applications where l(1) approximate distances may be preferred. Dimensionality reduction in l(1) often use Cauchy random projections that multiply the original data matrix B is an element of R-nxD with a Cauchy random matrix R is an element of R-kxn (k << n), resulting in a projected matrix C is an element of R-kxD. In this paper, an analogous of the Restricted Isometry Property for dimensionality reduction in l(1) is proposed using explicit tail bounds for the geometric mean of the random projections. A set of signal reconstruction algorithms from the Cauchy random projections are then developed given that the large suite of reconstruction algorithms developed in compressive sensing perform poorly due to the lack of finite second-order statistics in the projections. These algorithms are based on regularized coordinate-descent Myriad estimates using both l(0) and Lorentzian norms as sparsity inducing terms.