Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zimo Yan, Zheng Xie, Runfan Duan, Chang Liu, Wumei Du

TL;DR

This work interprets 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.

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.

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

TL;DR

This work interprets 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.

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.
Paper Structure (44 sections, 8 theorems, 107 equations, 4 figures, 3 tables)

This paper contains 44 sections, 8 theorems, 107 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Graph structure learning and locality limitations
  4. Spectral positional encodings
  5. Distance based positional encodings
  6. Graph based models for drug-drug interaction prediction
  7. Summary and remaining gap
  8. Preliminaries
  9. Notations
  10. Problem Statement
  11. Theoretical Analysis
  12. Assumptions
  13. Main Theoretical Results
  14. Auxiliary quantities.
  15. Implications for expressivity relative to NoPE
  16. ...and 29 more sections

Key Result

Theorem 2

Fix $t>0$ and $m\in\mathbb{N}$. Let $G=(V,E)$ be a finite connected graph and let $R\ge 1$. Let $\psi:[0,R]\to\mathbb{R}_+$ be strictly increasing. Define and Then, for all $u,v\in V$ with $\mathrm{SPD}(u,v)\le R$,

Figures (4)

  • Figure 1: Overview of spectral/diffusion and anchor-distance positional encodings, and the algebraic bridge developed in this work.
  • Figure 2: Proposed DDI framework: positional encodings augment molecular graphs, which are encoded by a shared multi-scale GNP and decoded via co-attention and relation-aware scoring.
  • Figure 3: DrugBank diffusion-geometry diagnostics (80 graphs).Top: normalized Frobenius gap vs. monotonicity residual $\Delta\hat{L}$ (median/p95 and Spearman $\rho$). Bottom: tail CCDF of $\log_{10}\kappa(K_{AA})$ and $\log_{10}\kappa(A)$.
  • Figure 4: Qualitative comparison of diffusion-based embeddings for a single DrugBank molecular graph (DB00006).

Theorems & Definitions (15)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • Remark 4: Jittered anchors are generic a.s.
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 1
  • Lemma 8: Li et al. li2009distance
  • Lemma 9
  • ...and 5 more