RECONSTRUCTION OF SPARSE SIGNALS FROM l1 DIMENSIONALITY-REDUCED CAUCHY RANDOM-PROJECTIONS

Dimensionality reduction via linear random projections are used in numerous applications including data streaming, information retrieval, data mining, 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) use Cauchy random projections that multiply the original data matrix B is an element of R(Dxn) with a Cauchy random matrix R is an element of R(nxk) (k << min(n, D)), resulting in a projected matrix C is an element of R(Dxk). This paper focuses on developing signal reconstruction algorithms from Cauchy random projections, where the large suite of reconstruction algorithms developed in compressive sensing perform poorly due to the lack of finite second-order statistics in the projections. In particular, a set of regularized coordinate-descent Myriad regression based reconstruction algorithms are developed using, both l(0) and Lorentzian norms as sparsity inducing terms. The l(0)-regularized algorithm shows superior performance to other standard approaches. Simulations illustrate and compare accuracy of reconstruction.