On real and complex valued ℓ ₁-norm minimization for overcomplete blind source separation

A maximum a-posteriori approach for overcomplete blind source separation based on Laplacian priors usually involves ℓ ₁ -norm minimization. It requires different approaches for real and complex numbers as they appear for example in the frequency domain. In this paper we compare a combinatorial approach for real numbers with a second order cone programming approach for complex numbers. Although the combinatorial solution with a proven minimum number of zeros is not theoretically justified for complex numbers, its performance quality is comparable to the performance of the second order cone programming (SOCP) solution. However, it has the advantage that it is faster for complex overcomplete BSS problems with low input/output dimensions.