Fast algorithms for reconstruction of sparse signals from cauchy random projections

Recent work on dimensionality reduction using Cauchy random projections has emerged for applications where ℓ₁ distance preservation is preferred. An original sparse signal b ϵ ℝⁿ is multiplied by a Cauchy random matrix R ϵ ℝ^n×k (k≪n), resulting in a projected vector c ϵ ℝ^k. Two approaches for fast recover of b from the Cauchy vector c are proposed. The two algorithms are based on a regularized coordinate-descent Myriad regression using both ℓ₀ and convex relaxation as sparsity inducing terms. The key element is to start, in the first iteration, by finding the optimal estimate value for each coordinate, and selectively updating only the coordinates with rapid descent in subsequent iterations. For the particular case of the ℓ₀ regularized approach, an approximation function for the ℓ₀-norm is given due to it is non-differentiable norm [1]. Performance comparisons of the proposed approaches to the original regularized coordinate-descent method are included.