Free denoising via overlap measures and c-freeness techniques

Abstract

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 $E(a|a+b)$. To that end, we study a probability measure ${\mu }_{a,a+b}^{(\mathrm{ov})}$ on ${\mathbb{R}}^2$ which we call the overlap measure. We show that ${\mu }_{a,a+b}^{(\mathrm{ov})}$ is absolutely continuous with respect to the product measure ${\mu }_a\times {\mu }_{a+b}$. The Radon-Nikodym derivative gives direct access to $E(a|a+b)$. We show that analogous results hold in the case of multiplicative noise when $a,b$ are positive and the aim is to find $E(a|a^{1/2}b{a}^{1/2})$. In a parallel development we show that, for a general selfadjoint expression $P(a,b)$ made with $a$ and $b$, finding $E(a|P(a,b))$ is equivalent to finding the distribution of $P(a,b)$ in a certain two-state probability space $(\mathcal{A},\phi ,\chi )$, 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.

Version 3: Framework of Borel non-commutative probability space introduced, relation to matrix denoising updated