Free convolution
In free probability, a mathematical theory developed only since about 1990, free deconvolution is a recent application to signal processing. It enables one to compute the eigenvalues of involved models of sum or product of random matrices using combinatorial techniques. It has some some strong connections with other works on G-estimation of Girko.
As a straightforward example, suppose that A and B are independent large square Hermitian (or symmetric) random matrices, then under some very general conditions, free deconvolution enables to:
- Deduce the eigenvalue distribution of A from those of A + B and B.
- Deduce the eigenvalue distribution of A from those of AB and B.
The concept is even broader as it provides a method to retrieve the eigenvalue distribution of A from any functional ƒ(A, B) and B(ƒ(A, B) is a function of the two matrices A and B).
The applications in wireless communications, finance and biology have provided a useful framework when the number of observations is of the same order as the dimensions of the system.
References
- "Free Deconvolution for Signal Processing Applications", O. Ryan and M. Debbah, ISIT 2007, pp. 1846–1850