Local Identifiability of Deep ReLU Neural Networks: the Theory

Joachim Bona-Pellissier, Francois Malgouyres, Francois Bachoc

Keywords: Deep Learning, ReLU networks, Conditions of identifiability, Lifting operator

Abstract: Is a sample rich enough to determine, at least locally, the parameters of a neural network? To answer this question, we introduce a new local parameterization of a given deep ReLU neural network by fixing the values of some of its weights. This allows us to define local lifting operators whose inverses are charts of a smooth manifold of a high dimensional space. The function implemented by the deep ReLU neural network composes the local lifting with a linear operator which depends on the sample. We derive from this convenient representation a geometrical necessary and sufficient condition of local identifiability. Looking at tangent spaces, the geometrical condition provides: 1/ a sharp and testable necessary condition of identifiability and 2/ a sharp and testable sufficient condition of local identifiability. The validity of the conditions can be tested numerically using backpropagation and matrix rank computations.

TL;DR: We characterize theoretically the question of local identifiability for deep ReLU neural networks and we provide numerically testable conditions.