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 $$ E (a \, | \, a+b)$$ E ( a | a + b ) . To that end, we study a probability measure $$ \mu ^{( \textrm{ov} )}_{a,a+b} $$ μ a , a + b ( ov ) on $$ \mathbb {R} ^2$$ R 2 which we call the overlap measure . We show that $$ \mu ^{( \textrm{ov} )}_{a,a+b} $$ μ a , a + b ( ov ) is absolutely continuous with respect to the product measure $$\mu _a\times \mu _{a+b}$$ μ a × μ 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 (