Track papers across labs, topics, and trends.
Search papers, labs, and topics across Lattice.
This paper addresses the challenge of recovering latent potentials from observed flow on directed graphs, which is typically ill-posed due to standard ridge regularization collapsing the recovered ordering. By introducing gauge-invariant graph Dirichlet energy, the authors achieve parameter-insensitivity and stability across a wide range of regularization parameters, significantly outperforming ridge regularization in preserving dynamic range. The method not only retains 28-41% of the interior dynamic range on multiple datasets but also has implications for improving graph neural networks by preventing oversmoothing in deep directed graph convolutional networks (GCNs).
Gauge-invariant regularization can retain up to 41% of dynamic range in potential recovery, while traditional methods collapse to as little as 0.2%.
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.