发表机构
Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有向图谱位置编码面临的障碍，提出可学习谱PEs形式\(h_\theta(A_q)\,R\)，通过厄米特块克里洛夫子空间计算，证明其规范不变性及相关性质，在有向SBM等实验中展现优势，优于无PE和多项式基线。
AI 中文摘要
有向图的谱位置编码（PEs）面临两个障碍：磁拉普拉斯算子每个势需要\(O(n^3)\)的厄米特特征分解，且其复特征向量仅在酉规范下定义，此前工作通过基不变架构处理。我们提出形式为\(h_\theta(A_q)\,R\)的可学习谱PEs，其中\(A_q\)是归一化磁算子，\(h_\theta\)是可学习标量谱响应，\(R\)是随机探针块。因其是算子的矩阵函数，所以本质上是规范不变的。我们仅通过稀疏矩阵 - 向量乘积在厄米特块克里洛夫子空间中计算它，证明\(k = O(\log(1/\varepsilon))\)个块步骤对热 - 预解式响应族一致足够，并给出一个覆盖数论证，解释为何低维结构化族能泛化而自由的每个特征值权重会过拟合。在一个构造上对称化无信息的有向SBM上，随着深度增加，方向盲PEs保持随机，而磁克里洛夫PEs收敛到精确特征分解预言机。相同探针产生规范不变的成对特征，其蒙特卡罗误差为\(1/\sqrt{s}\)，无向\(q = 0\)情况在异质基准上优于无PE和多项式基线。
英文摘要
Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_θ(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_θ$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \emph{matrix function} of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.
Comments8 pages main, theorem and type fixed
