Eigenbasis-Independent Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces
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...
| Authors | Jiaqing Xie, Yuxin Wang |
|---|---|
| Primary category | cs.LG |
| Published | 8 July 2026 |
| Last updated at source | 11 July 2026 |
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