Graph Residual Conjugate Diffusion: SNR-Equalized Heat Flow for Graph Signals

2026-09-30

AI Summary

This study addresses the inefficient sampling in high-noise regions caused by uneven spectral energy distributions in graph signals under isotropic noise. To this end, it proposes the GRCD model, which introduces mode-dependent clocks to equalize signal-to-noise ratios (SNR). By combining a finite terminal SNR reference distribution with an exact modal propagator, the method analytically propagates Gaussian components and numerically integrates residual scores, enabling efficient generation via probability flow ODEs. Evaluated on five datasets including METR-LA, GRCD reduces the maximum mean discrepancy by 22 to 36 times compared to the strongest baseline while decreasing sampling time by 87%.

Abstract

Diffusion models generate data by reversing a forward corruption process that typically approaches a simple Gaussian prior. Recent work has extended this framework to signals supported on fixed graphs, e.g., road-network traffic and sensor-network measurements. Many graph signals have nonuniform spectral energy, whereas isotropic corruption adds the same conditional noise variance to every graph-frequency mode. Driving all modes to near-zero terminal signal-to-noise ratio (SNR) requires strong corruption, which increases the noise range that must be covered under a fixed sampling budget. We introduce Graph Residual Conjugate Diffusion (GRCD), which replaces the shared clock of graph heat diffusion with a mode-dependent clock that gives every graph-Fourier mode the same conditional SNR. GRCD fits a zero-mean graph-spectral Gaussian reference on the training split and stops at a finite terminal SNR at which the propagated reference still carries the fitted spectral variances. The Gaussian component has an exact modewise propagator in the probability-flow ODE, so sampling advances it analytically and integrates only the learned residual score numerically. We evaluate GRCD on five settings (METR-LA traffic, Molene weather, and three stochastic block models) against seven comparators under a matched protocol: Graph-Aware Diffusion (GAD), EDM (graph backbone), two adaptations of Whitened Score Diffusion (WSD), and three preconditioning controls. At four function evaluations (NFEs), GRCD lowers averaged maximum mean discrepancy (aMMD) by 22 to 36 times over the best comparator on all five settings, reaching 0.054 on METR-LA, where it clears an aMMD 0.1 target with 87% less sampling wall-clock time than the cheapest comparator that reaches it. Fitting the terminal reference reduces aMMD by 2.7 to 7.3 times at finite terminal SNR, while the factors shrink to 1.00 to 1.01 near zero.

Problem

• Graph signal generation
• Diffusion models
• Nonuniform spectral energy
• Signal-to-noise ratio
• Isotropic corruption

Innovation

• Graph Residual Conjugate Diffusion
• SNR equalization
• mode-dependent clock
• probability-flow ODE
• residual score