Local Identifiability of Deep ReLU Neural Networks: the Theory

Joachim Bona-Pellissier · Fran├žois Malgouyres · Francois Bachoc

Hall J #814

Keywords: [ ReLU networks ] [ Lifting operator ] [ Conditions of identifiability ] [ Deep Learning ]

[ Abstract ]
[ Paper [ Poster [ OpenReview
Wed 30 Nov 9 a.m. PST — 11 a.m. PST


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.

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