  We present a method to reconstruct autocorrelated signals together with their
autocorrelation structure from nonlinear, noisy measurements for arbitrary
monotonous nonlinear instrument response. In the presented formulation the
algorithm provides a significant speedup compared to prior implementations,
allowing for a wider range of application. The nonlinearity can be used to
model instrument characteristics or to enforce properties on the underlying
signal, such as positivity. Uncertainties on any posterior quantities can be
provided due to independent samples from an approximate posterior distribution.
We demonstrate the methods applicability via simulated and real measurements,
using different measurement instruments, nonlinearities and dimensionality.
