{
  "abstract": "Recovering a latent potential from observed flow on a directed graph (a discrete Poisson problem with Dirichlet boundaries) is ill-posed, and the standard fix backfires: ridge regularization shrinks toward a gauge-meaningless origin, collapsing and reversing the recovered ordering ($+0.81\\to-0.42$ rank correlation against a planted ground truth). The gauge-invariant graph Dirichlet energy removes the hazard and delivers parameter-insensitivity: the estimate is stable across four orders of magnitude in $λ$, whereas ridge inverts the ordering for every $λ>0$. We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone via a Poisson residual. The $H^1$ seminorm is classical; what is new is the gauge diagnosis, the parameter-insensitivity it buys, and an ablation showing the result is robust to the extraction method. On three public clickstream corpora the gauge-invariant estimate retains $28$--$41\\%$ of the interior dynamic range while ridge collapses to as little as $0.2\\%$. The same gauge invariance carries into graph neural networks -- neutralizing the constant mode per layer prevents the oversmoothing that collapses a deep directed GCN -- linking this classical inverse problem to a central question in graph learning.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13609",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad Forouhesh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7a5gt5d4jrk8h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13609",
  "fulltext_url": null,
  "id": "7a5gt5d4jrk8h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13609",
  "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.IR"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7168755106",
  "pdf_url": "https://arxiv.org/pdf/2607.13609",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7a5gt5d4jrk8h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:42Z",
      "native_id": "2607.13609",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13609"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs",
  "title_text": "Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:42Z",
  "url": "https://research.draconitic.com/paper/7a5gt5d4jrk8h",
  "venue": "arXiv",
  "versions": []
}