发表机构
Amirkabir University of Technology（伊朗阿米尔卡比尔理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究从有向图流中恢复潜在势的问题，提出规范不变的图狄利克雷能量方法，该方法具有参数不敏感性，能保留动态范围，定位吸收边界，并通过实验验证其在点击流语料库上的优势，还将其与图神经网络联系起来。
AI 中文摘要
从有向图上观测到的流中恢复潜在势（具有狄利克雷边界的离散泊松问题）是不适定的，标准的岭正则化会朝着规范无意义的原点收缩，导致恢复的排序崩溃和反转（与植入的真实情况的秩相关性为+0.81→-0.42）。规范不变的图狄利克雷能量消除了这种风险并实现了参数不敏感性：估计在λ的四个数量级上是稳定的，而岭正则化对于每个λ>0都会反转排序。我们证明了简化求解是对称正定的，并且在岭正则化使其崩溃的地方精确地保留了动态范围，并且仅通过泊松残差就可以从流中定位吸收边界。H^1半范数是经典的；新的是规范诊断、它带来的参数不敏感性以及消融实验表明结果对提取方法具有鲁棒性。在三个公共点击流语料库上，规范不变估计保留了28%-41%的内部动态范围，而岭正则化则缩小到低至0.2%。相同规范不变性带入图神经网络——中和每层的常数模式可防止深度有向图卷积网络崩溃的过度平滑，将这个经典反问题与图学习中的一个核心问题联系起来。
英文摘要
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.
Comments17 pages, 6 figures, submitted to LoG 2026
