Reconstruction of Sparse Signals From $\ell _1$ 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 ℓ₂ distance preservation, a large body of work has emerged for applications where ℓ₁ approximate distances may be preferred. Dimensionality reduction in ℓ₁ often use Cauchy random projections that multiply the original data matrix B ∈ R^n×D with a Cauchy random matrix R ∈^k×n (k≪n), resulting in a projected matrix C ∈^k×D. In this paper, an analogous of the Restricted Isometry Property for dimensionality reduction in is ℓ₁ 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 ℓ₀ and Lorentzian norms as sparsity inducing terms.