ON-LINE QLunch: Ralf Müller

Consider a linear random channel y = Hx + n with H biunitarily invariant and the components of x independent identically distributed (iid) and n iid Gaussian. Let d = f(y|H) be an estimator for x. Under mild conditions on this estimator, there exists an equivalent scalar channel y’ = x’ + n’ and an estimator d’ = f'(y) with the same joint statistics between estimate and true data, i.e., Pr(d_k,x_k) = Pr(d’,x’) for all components 0<k<K in the limit K to infinity.

The noise n’ of the equivalent scalar channel is Gaussian and independent of x', if the spin glas formulation of the estimation problem is replica symmetric. If it breaks replica symmetry, the equivalent scalar noise is non-Gaussian and data-dependent. Its joint distribution with the data x’ can be characterized for an arbitrary number of breaking steps.