aiqbench analysis
This paper addresses a critical instability in recovering latent potentials from flow on directed graphs by introducing a gauge-invariant regularization method that preserves dynamic range and ordering, unlike standard ridge regularization which collapses these metrics. The approach demonstrates robustness across orders of magnitude in hyperparameters and directly mitigates oversmoothing issues in deep graph neural networks, bridging classical inverse problems with modern graph learning architectures.
Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
arXiv:2607.13609v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Information Retrieval (cs.IR); Signal Processing (eess.SP); Machine Learning (stat.ML)
arXiv:2607.13609 [cs.LG]
(or arXiv:2607.13609v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.13609 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 293 words
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