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Multilayer (or deep) networks are powerful probabilistic models based onmultiple stages of a linear transform followed by a non-linear (possiblyrandom) function. In general, the linear transforms are defined by matrices andthe non-linear functions are defined by information channels. These models havegained great popularity due to their ability to characterize complexprobabilistic relationships arising in a wide variety of inference problems.The contribution of this paper is a new method for analyzing the fundamentallimits of statistical inference in settings where the model is known. Thevalidity of our method can be established in a number of settings and isconjectured to hold more generally. A key assumption made throughout is thatthe matrices are drawn randomly from orthogonally invariant distributions.  Our method yields explicit formulas for 1) the mutual information; 2) theminimum mean-squared error (MMSE); 3) the existence and locations of certainphase-transitions with respect to the problem parameters; and 4) the stationarypoints for the state evolution of approximate message passing algorithms. Whenapplied to the special case of models with multivariate Gaussian channels ourmethod is rigorous and has close connections to free probability theory forrandom matrices. When applied to the general case of non-Gaussian channels, ourmethod provides a simple alternative to the replica method from statisticalphysics. A key observation is that the combined effects of the individualcomponents in the model (namely the matrices and the channels) are additivewhen viewed in a certain transform domain.

Additivity of Information in Multilayer Networks via Additive Gaussian Noise Transforms