arXiv：cs.LG· Mohammad Forouhesh·· 3 小时前
规范不变、参数不敏感的正则化：从有向图流中恢复势函数
Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
AI 导读
针对有向图流恢复潜在势函数的病态问题，研究者提出规范不变的图 Dirichlet 能量正则化，在 λ 跨越四个数量级时估计保持稳定，而 ridge 对任意 λ>0 都会反转排序。在三个公开点击流语料上，该方法保留 28%–41% 的内部动态范围，ridge 则低至 0.2%。该规范不变性还可用于图神经网络，逐层中和常数模态可防止深度有向 GCN 的过平滑。
正文
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.
| Comments: | 19 pages, 6 figures, Proceedings of the Fifth Learning on Graphs Conference (LoG 2026), PMLR 338 |
| Subjects: | Machine Learning (cs.LG); Information Retrieval (cs.IR); Signal Processing (eess.SP); Machine Learning (stat.ML) |
| Cite as: | arXiv:2607.13609 [cs.LG] |
| (or arXiv:2607.13609v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2607.13609 arXiv-issued DOI via DataCite |
Submission history
From: Mohammad Hossein Forouhesh Tehrani [view email]
[v1]
Wed, 15 Jul 2026 08:56:57 UTC (123 KB)
[v2]
Wed, 14 Oct 2026 07:00:30 UTC (70 KB)
来源：arXiv：cs.LG · arxiv.org