Bayesian Inference with Nonlinear Generative Models: Comments on Secure Learning
This work addresses secure communication for applications like data privacy, though it appears incremental as it builds on existing replica method techniques.
The paper tackles the problem of secure learning with nonlinear generative models, showing that they exhibit an all-or-nothing phase transition in Bayesian inference and can achieve the secrecy capacity of wiretap channels without additional coding.
Unlike the classical linear model, nonlinear generative models have been addressed sparsely in the literature of statistical learning. This work aims to bringing attention to these models and their secrecy potential. To this end, we invoke the replica method to derive the asymptotic normalized cross entropy in an inverse probability problem whose generative model is described by a Gaussian random field with a generic covariance function. Our derivations further demonstrate the asymptotic statistical decoupling of the Bayesian estimator and specify the decoupled setting for a given nonlinear model. The replica solution depicts that strictly nonlinear models establish an all-or-nothing phase transition: There exists a critical load at which the optimal Bayesian inference changes from perfect to an uncorrelated learning. Based on this finding, we design a new secure coding scheme which achieves the secrecy capacity of the wiretap channel. This interesting result implies that strictly nonlinear generative models are perfectly secured without any secure coding. We justify this latter statement through the analysis of an illustrative model for perfectly secure and reliable inference.