Estimating uncertainty from feed-forward network based sensing using quasi-linear approximation

A fundamental problem in neural network theory is the quantification of uncertainty as it propagates through these constructs. Such quantification is crucial as neural networks become integrated into broader engineered systems that render decisions based on their outputs. In this paper, we engage the problem of estimating uncertainty in feedforward neural network constructs. Mathematically, the problem, in essence, amounts to understanding how the moments of an input distribution become modifies as they move through network layers. Despite its straightforward formulation, the nonlinear nature of modern feedforward architectures makes this is a mathematically challenging problem. Most contemporary approaches rely on some form of Monte Carlo sampling to construct inter-laminar distributions. Here, we borrow an approach from the control systems community known as quasilinear approximation, to enable a more analytical approach to the uncertainty quantification problem in this setting. Specifically, by using quasilinear approximation, nonlinearities are linearized in terms of the expectation of their gain in an input-output sense. We derive these expectations for several commonly used nonlinearities, under the assumption of Gaussian inputs. We then establish that the ensuing approximation is accurate relative to traditional linearization. Furthermore, we provide a rigorous example how this method can enable formal estimation of uncertainty in latent variables upstream of the network, within a target-tracking case study.