Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs

arXiv:2607.13609 · cs.LG, cs.IR, eess.SP, stat.ML · Submitted 2026-07-15 · Read on arXiv

cs.LG, cs.IR, eess.SP, stat.ML

Submitted: 2026-07-15

Updated: 2026-10-14

Comments: 17 pages, 6 figures, submitted to LoG 2026

Code: https://github.com/MohammadForouhesh/gauge-flow-recovery

License: http://creativecommons.org/licenses/by/4.0/

The gist: 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

Terminology

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 lambda, whereas ridge inverts the ordering for every lambda>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.

Related papers