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