Bridging Distance and Spectral Positional Encodings via Anchor-Based Diffusion Geometry Approximation

TL;DR

This paper explores the relationship between spectral and anchor-based distance encodings in molecular graph learning, proposing a diffusion geometry approximation that improves encoding effectiveness.

Contribution

It introduces a novel interpretation of distance encodings as low-rank surrogates of diffusion geometry and derives an explicit trilateration map with theoretical guarantees.

Findings

Distance-driven Nyström scheme effectively recovers diffusion geometry.

Both Laplacian and distance encodings outperform no-encoding baselines.

The approach is validated on DrugBank molecular graphs with improved prediction performance.

Abstract

Molecular graph learning benefits from positional signals that capture both local neighborhoods and global topology. Two widely used families are spectral encodings derived from Laplacian or diffusion operators and anchor-based distance encodings built from shortest-path information, yet their precise relationship is poorly understood. We interpret distance encodings as a low-rank surrogate of diffusion geometry and derive an explicit trilateration map that reconstructs truncated diffusion coordinates from transformed anchor distances and anchor spectral positions, with pointwise and Frobenius-gap guarantees on random regular graphs. On DrugBank molecular graphs using a shared GNP-based DDI prediction backbone, a distance-driven Nystr\"om scheme closely recovers diffusion geometry, and both Laplacian and distance encodings substantially outperform a no-encoding baseline.