  Multilayer (or deep) networks are powerful probabilistic models based on
multiple stages of a linear transform followed by a non-linear (possibly
random) function. In general, the linear transforms are defined by matrices and
the non-linear functions are defined by information channels. These models have
gained great popularity due to their ability to characterize complex
probabilistic relationships arising in a wide variety of inference problems.
The contribution of this paper is a new method for analyzing the fundamental
limits of statistical inference in settings where the model is known. The
validity of our method can be established in a number of settings and is
conjectured to hold more generally. A key assumption made throughout is that
the matrices are drawn randomly from orthogonally invariant distributions.
  Our method yields explicit formulas for 1) the mutual information; 2) the
minimum mean-squared error (MMSE); 3) the existence and locations of certain
phase-transitions with respect to the problem parameters; and 4) the stationary
points for the state evolution of approximate message passing algorithms. When
applied to the special case of models with multivariate Gaussian channels our
method is rigorous and has close connections to free probability theory for
random matrices. When applied to the general case of non-Gaussian channels, our
method provides a simple alternative to the replica method from statistical
physics. A key observation is that the combined effects of the individual
components in the model (namely the matrices and the channels) are additive
when viewed in a certain transform domain.
