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

arXiv: https://arxiv.org/abs/2609.39658 · HTML: https://mlsanity.com/paper/2609.39658/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 21 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [PriorGrad: Improving Conditional Denoising Diffusion Models with Data-Dependent Adaptive Prior](https://mlsanity.com/paper/2106.06406/) (2021)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Probabilistic Mass Mapping with Neural Score Estimation](https://mlsanity.com/paper/2201.05561/) (2022)
- [Graph-Aware Diffusion for Signal Generation](https://mlsanity.com/paper/2510.05036/) (2025)
- [Whitened Score Diffusion: A Structured Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2505.10311/) (2025)

## Abstract (arXiv)

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.