Relevance Propagation as Entropic Optimal Transport on Filippov Neural Networks: Sinkhorn Propagation and Region Relevance in Image Classifiers

Olivier Gibaru, Eric Nyiri

Relevance propagation, which redistributes a network's prediction backward through the layers, is usually treated as a heuristic. We show that it is the solution of a single variational problem: Layer-wise Relevance Propagation (LRP) is the divergence-free flow on the active computational graph of zero relative entropy to the backward transition law of the forward activation flow; for a non-negative reference, this is a relative entropy between Markov measures on backward paths. The rules LRP$_0$, LRP$_{z^+}$ and LRP$_{α,β}$ follow from this principle by the choice of a reference, and are instances of one rule, LRPN$_{α,β}$, whose weights may vary from node to node, for instance with the node's activity, in the spirit of Hebb's principle. The principle rests on a measure-theoretic formalization of feed-forward networks as directed acyclic graphs of Filippov operators, which covers piecewise-smooth and discontinuous architectures (e.g. hard-routed Mixture-of-Experts) with a Clarke-Filippov generalized-gradient calculus; it also characterizes backpropagation as the adjoint equation of a Lagrangian relaxation. On networks of ReLU and max-pooling nodes, LRP$_0$ is…