Abstract
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$_{\alpha,\beta}$ follow from this principle by the choice of a reference, and are instances of one rule, LRPN$_{\alpha,\beta}$, 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 Gradient$\times$Input, with an explicit jump across a kink where LRP$_{z^+}$ stays continuous. Taking the entropy edge by edge yields a new rule, Sinkhorn Relevance Propagation. Aggregated over an input region, the flow of a non-negative reference tells, with exact accounting, which hidden units carry its evidence, the conditioned flow being a Schrödinger bridge. On VGG-16, the conservation identities hold to round-off, and the rules are told apart by the sign of their maps, the share of the output left to the bias nodes, faithfulness, localization and class dependence; on ConvNeXt-Tiny, LRPN$_{\alpha,\beta}$ conserves the output and its Hebbian weights give the highest ABPC of the rules tested.
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URL
https://arxiv.org/abs/2611.05594
