{
  "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$_{α,β}$ 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 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$_{α,β}$ conserves the output and its Hebbian weights give the highest ABPC of the rules tested.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05594",
  "author_total": 2,
  "authors": [
    {
      "name": "Olivier Gibaru"
    },
    {
      "name": "Eric Nyiri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xy7c2a31nwgb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05594",
  "fulltext_url": null,
  "id": "0xy7c2a31nwgb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05594",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CV"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05594",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xy7c2a31nwgb.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05594",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Relevance Propagation as Entropic Optimal Transport on Filippov Neural Networks: Sinkhorn Propagation and Region Relevance in Image Classifiers",
  "title_text": "Relevance Propagation as Entropic Optimal Transport on Filippov Neural Networks: Sinkhorn Propagation and Region Relevance in Image Classifiers",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/0xy7c2a31nwgb",
  "venue": "arXiv",
  "versions": []
}