Free denoising via overlap measures and c-freeness techniques

We study the problem of free denoising. For free selfadjoint random variables a, b, where we interpret a as a signal and b as noise, we find $E\left(a\right|a+b)$ . To that end, we study a probability measure ${\mu }_{a,a+b}^{\left(\text{ov}\right)}$ on $R^2$ which we call the overlap measure. We show that ${\mu }_{a,a+b}^{\left(\text{ov}\right)}$ is absolutely continuous with respect to the product measure ${\mu }_a\times {\mu }_{a+b}$ . The Radon-Nikodym derivative gives direct access to $E\left(a\right|a+b)$ . We show that analogous results hold in the case of multiplicative noise when a, b are positive and the aim is to find $E\left(a\right|a^{1/2}b{a}^{1/2})$ . In a parallel development we show that, for a general selfadjoint expression P(a, b) made with a and b, finding $E\left(a\right|P(a,b))$ is equivalent to finding the distribution of P(a, b) in a certain two-state probability space $(A,\phi ,\chi )$ , where a, b are c-free with respect to $(\phi ,\chi )$ in the sense of Bożejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract $W^{\ast }$ -probability space) relates to the notion of "matrix denoising" previously discussed in the random matrix literature.