Free Deconvolution: from Theory to Practice

In this paper, we provide an algorithmic method to compute the singular values of sum of rectangular matrices based on the free cumulants approach and illustrate its application to wireless communications. We first recall the algorithms working for sum/products of square random matrices, which have already been presented in some previous papers and we then introduce the main contribution of this paper which provides a general method working for rectangular random matrices, based on the recent theoretical work of Benaych-Georges. In its full generality, the computation of the eigenvalues requires some sophisticated tools related to free probability and the explicit spectrum (eigenvalue distribution) of the matrices can hardly be obtained (except for some trivial cases). From an implementation perspective, this has led the community to the misconception that free probability has no practical application. This contribution takes the opposite view and shows how the free cumulants approach in free probability provides the right shift from theory to practice.