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

• URL: http://arxiv.org/abs/2601.04517v1
• Date: Thu, 08 Jan 2026 02:31:03 GMT
• Title: Bridging Distance and Spectral Positional Encodings via Anchor-Based Diffusion Geometry Approximation
• Abstract summary: Two widely used families are spectral encodings derived from Laplacian or diffusion operators and anchor-based distance encodings built from shortest-path information.<n>We derive an explicit trilateration map that reconstructs truncated diffusion coordinates from transformed anchor distances and anchor spectral positions.<n>On DrugBank molecular graphs using a shared GNP-based DDI prediction backbone, a distance-driven Nystrm scheme closely recovers diffusion geometry.
• Score: 7.270459013324939
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 Nyström scheme closely recovers diffusion geometry, and both Laplacian and distance encodings substantially outperform a no-encoding baseline.

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