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

Jinwei Li; Daniel Tenbrinck

cs.LGSep 30, 2026

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

Authors: Jinwei Li, Daniel Tenbrinck

Organizations: Department of Data Science Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen, Germany

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.

Figures & tables

METHOD	NFE 4	NFE 8	NFE 16	NFE 64
EDM	$2.7757\pm 0.0040$	$1.3996\pm 0.0605$	$0.2640\pm 0.0139$	$0.1155\pm 0.0053$
GAD	$2.9091\pm 0.0016$	$2.9093\pm 0.0056$	$2.8878\pm 0.0146$	$2.8785\pm 0.0163$
WSD-style, scalar	$1.8752\pm 0.2182$	$0.3941\pm 0.1355$	$0.2695\pm 0.0587$	$0.2572\pm 0.0183$
WSD-style, graph prior	$1.2431\pm 0.2440$	$0.2410\pm 0.1347$	$0.1980\pm 0.0928$	$0.0966\pm 0.0325$
Isotropic VE	$2.7581\pm 0.0084$	$0.7810\pm 0.0240$	$0.1816\pm 0.0139$	$0.1152\pm 0.0087$
Static precond.	$2.7728\pm 0.0088$	$0.7993\pm 0.0155$	$0.1755\pm 0.0070$	$0.1134\pm 0.0043$

Table 1: Matched comparison on METR-LA. aMMD at the validation-converged stage and a fixed nominal NFE budget (Appendix A.6.5 ). Lower numbers are better and bold denotes the best mean in each column. Results are mean

\pm

sample standard deviation over three random seeds. The last two rows use the exponential-residual solver.

Panel of Figure 1 — Figure 1: Terminal-reference effects are gated by terminal SNR. Left: Seed-paired scalar-to-fitted aMMD ratio at NFE $16$ across five graph settings. The dashed rule at $1$ marks no effect and values above it favor the fitted reference. Right: Per-sensor standard deviation on METR-LA for one of the three random seeds with each dot placed at its sensor’s longitude and latitude. Color is the generated standard deviation relative to the correct value of $1.0$ . Lower color contrast is better.

CONFIGURATION	NFE 8	NFE 12	NFE 16	NFE 64
Mode-dependent clock	$0.4205\pm 0.0085$	$0.3817\pm 0.0067$	$0.3722\pm 0.0069$	$0.3632\pm 0.0076$
+ fitted terminal ref.	$0.0603\pm 0.0029$	$0.0530\pm 0.0023$	$0.0513\pm 0.0021$	$\mathbf{0.0499\pm 0.0018}$
+ residual score	$0.0622\pm 0.0028$	$0.0549\pm 0.0034$	$0.0537\pm 0.0034$	$0.0531\pm 0.0033$
+ exp. solver	$\mathbf{0.0519\pm 0.0057}$	$\mathbf{0.0493\pm 0.0036}$	$\mathbf{0.0499\pm 0.0032}$	$0.0527\pm 0.0032$

Table 2: Ablation ladder on METR-LA. aMMD at

\kappa=2

, validation-converged stage, mean

\pm

s.d. over three random seeds. Each row adds one component to the row above. Lower numbers are better and bold denotes the best mean in each column. All rows use the uniform grid in

t

. The first and third rows are ablation-only configurations.

SOLVER	NFE 4	NFE 8	NFE 16	NFE 32
Heun	$0.1608\pm 0.0205$	$0.0606\pm 0.0014$	$0.0514\pm 0.0027$	$0.0523\pm 0.0029$
DPM 2M, uniform $\ell$	$0.0659\pm 0.0028$	$0.0523\pm 0.0022$	$0.0509\pm 0.0022$	$0.0522\pm 0.0029$
DPM 2M, $\rho{=}3$	$\mathbf{0.0523\pm 0.0010}$	$0.0516\pm 0.0020$	$0.0507\pm 0.0026$	$0.0522\pm 0.0029$
DPM 3M, uniform $\ell$	$0.0750\pm 0.0023$	$0.0552\pm 0.0021$	$0.0516\pm 0.0021$	$0.0524\pm 0.0029$
DPM 3M, $\rho{=}3$	$0.0708\pm 0.0023$	$0.0539\pm 0.0022$	$0.0510\pm 0.0023$	$0.0523\pm 0.0029$
Exp. residual ( $\rho{=}3$ , ours)	$0.0552\pm 0.0024$	$\mathbf{0.0488\pm 0.0028}$	$\mathbf{0.0497\pm 0.0020}$	$\mathbf{0.0519\pm 0.0029}$

Table 3: Solver comparison. aMMD on METR-LA, mean

\pm

s.d. over three random seeds. Lower numbers are better and bold marks the best mean at each NFE. DPM denotes DPM-Solver++.

Figure 2: Wall-clock sampling efficiency on METR-LA. aMMD versus measured seconds per

1000

generated samples. Lower and left are better.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

SETTING	SOURCE	$N$	TRAIN	VAL	TEST
METR-LA	traffic, $5$ -min ( Li et al., 2018 )	$207$	$20{,}000$	$3{,}000$	$5{,}000$
Molene	temperature ( Girault, 2015 )	$37$	$520$	$74$	$150$
SBM ( $c{=}1$ )	synthetic	$32$	$4{,}000$	$1{,}000$	$5{,}000$
SBM ( $c{=}4$ )	synthetic	$32$	$4{,}000$	$1{,}000$	$5{,}000$
SBM ( $c{=}16$ )	synthetic	$32$	$4{,}000$	$1{,}000$	$5{,}000$

Appendix

Table 4: Datasets.

N

denotes the number of graph nodes. Split sizes are the numbers of graph signals used in each experiment.

GRID	NFE 4	NFE 6	NFE 8	NFE 12	NFE 16
$\rho=1$	$0.5298\pm 0.1303$	$0.0876\pm 0.0257$	$0.0519\pm 0.0057$	$0.0493\pm 0.0036$	$0.0499\pm 0.0032$
$\rho=2$	$0.0702\pm 0.0127$	$0.0482\pm 0.0019$	$0.0498\pm 0.0030$	$0.0522\pm 0.0015$	$0.0508\pm 0.0016$
$\rho=3$	$0.0538\pm 0.0037$	$0.0484\pm 0.0011$	$0.0503\pm 0.0028$	$0.0525\pm 0.0015$	$0.0509\pm 0.0015$
$\rho=5$	$0.0558\pm 0.0028$	$0.0482\pm 0.0006$	$0.0504\pm 0.0027$	$0.0525\pm 0.0016$	$0.0509\pm 0.0015$
$\rho=7$	$0.0589\pm 0.0031$	$0.0482\pm 0.0006$	$0.0502\pm 0.0026$	$0.0524\pm 0.0016$	$0.0508\pm 0.0015$

Appendix

Table 5: Time-grid sweep for the exponential-residual solver on METR-LA. Test aMMD at

\kappa=2

, converged, mean

\pm

standard deviation over three random seeds.

GROUP	PARAMETER	VALUE
data	train / val / test caps	$20{,}000/3{,}000/5{,}000$
	normalization	per-node z-score, training statistics
graph	symmetrizer	$\tfrac{1}{2}({\bm{W}}+{\bm{W}}^{\top})$
	self-loops	removed after symmetrization
	Laplacian (model basis)	symmetric normalized
	Laplacian (evaluation)	combinatorial

Appendix

Table 6: Experimental hyperparameters. Values are shared across methods and datasets unless stated otherwise.

METHOD	NFE 4	NFE 8	NFE 16	NFE 64
EDM	$2.6268\pm 0.0378$	$1.3517\pm 0.0869$	$\mathbf{0.0951\pm 0.0110}$	$0.2126\pm 0.0330$
GAD	$2.7836\pm 0.0116$	$2.7613\pm 0.0558$	$2.7147\pm 0.0155$	$2.7303\pm 0.0261$
WSD-style, scalar	$2.6712\pm 0.0060$	$1.5278\pm 0.0730$	$1.1318\pm 0.0384$	$1.0181\pm 0.0134$
WSD-style, graph prior	$2.5191\pm 0.0101$	$0.8458\pm 0.0235$	$1.4101\pm 0.0507$	$1.1717\pm 0.0493$
Isotropic VE	$2.6264\pm 0.0319$	$0.5675\pm 0.0779$	$0.1243\pm 0.0304$	$0.2159\pm 0.0354$
Static precond.	$2.6326\pm 0.0287$	$0.7461\pm 0.0474$	$0.1325\pm 0.0163$	$0.2208\pm 0.0209$

Appendix

Table 7: Matched comparison on Molene. aMMD at the validation-converged stage and a fixed nominal NFE budget (Appendix A.6.5 ). Lower is better and bold denotes the best mean in each column. Results are mean

\pm

sample standard deviation over three random seeds.

METHOD	NFE 4	NFE 8	NFE 16	NFE 64
EDM	$1.7509\pm 0.0082$	$1.5238\pm 0.0101$	$0.4285\pm 0.0026$	$0.0119\pm 0.0017$
GAD	$2.2376\pm 0.0018$	$2.0090\pm 0.0085$	$1.9618\pm 0.0034$	$1.9395\pm 0.0078$
WSD-style, scalar	$1.3046\pm 0.1772$	$1.4082\pm 0.0704$	$1.3388\pm 0.0665$	$0.8715\pm 0.0535$
WSD-style, graph prior	$1.2084\pm 0.2054$	$1.4998\pm 0.1438$	$1.4533\pm 0.0949$	$0.8845\pm 0.0644$
Isotropic VE	$1.7524\pm 0.0067$	$1.2200\pm 0.0027$	$0.1704\pm 0.0014$	$\mathbf{0.0113\pm 0.0019}$
Static precond.	$1.7522\pm 0.0083$	$1.2339\pm 0.0152$	$0.1465\pm 0.0255$	$0.0210\pm 0.0073$

Appendix

Table 8: Matched comparison on SBM (

c{=}1

). aMMD at the validation-converged stage and a fixed nominal NFE budget (Appendix A.6.5 ). Lower is better and bold denotes the best mean in each column. Results are mean

\pm

sample standard deviation over three random seeds.

METHOD	NFE 4	NFE 8	NFE 16	NFE 64
EDM	$1.9471\pm 0.0082$	$1.6387\pm 0.0103$	$0.4467\pm 0.0045$	$0.0161\pm 0.0028$
GAD	$2.3262\pm 0.0017$	$2.1619\pm 0.0091$	$2.1281\pm 0.0028$	$2.1100\pm 0.0066$
WSD-style, scalar	$1.5867\pm 0.2069$	$1.3181\pm 0.1055$	$1.4083\pm 0.1201$	$0.9278\pm 0.0792$
WSD-style, graph prior	$1.4519\pm 0.2599$	$1.3647\pm 0.2030$	$1.5479\pm 0.1677$	$0.9285\pm 0.1027$
Isotropic VE	$1.9492\pm 0.0062$	$1.2787\pm 0.0074$	$0.1797\pm 0.0081$	$0.0164\pm 0.0041$
Static precond.	$1.9518\pm 0.0061$	$1.3265\pm 0.0086$	$0.1766\pm 0.0115$	$\mathbf{0.0132\pm 0.0035}$

Appendix

Table 9: Matched comparison on SBM (

c{=}4

). aMMD at the validation-converged stage and a fixed nominal NFE budget (Appendix A.6.5 ). Lower is better and bold denotes the best mean in each column. Results are mean

\pm

sample standard deviation over three random seeds.

METHOD	NFE 4	NFE 8	NFE 16	NFE 64
EDM	$2.1231\pm 0.0086$	$1.7378\pm 0.0141$	$0.4586\pm 0.0183$	$0.0204\pm 0.0049$
GAD	$2.4178\pm 0.0017$	$2.2966\pm 0.0091$	$2.2726\pm 0.0023$	$2.2574\pm 0.0064$
WSD-style, scalar	$1.5899\pm 0.4909$	$1.6443\pm 0.5671$	$1.4122\pm 0.2295$	$0.8368\pm 0.2536$
WSD-style, graph prior	$1.5082\pm 0.4521$	$1.7000\pm 0.7449$	$1.6301\pm 0.2184$	$0.8894\pm 0.1747$
Isotropic VE	$2.1257\pm 0.0056$	$1.3307\pm 0.0081$	$0.1824\pm 0.0181$	$0.0202\pm 0.0054$
Static precond.	$2.1269\pm 0.0058$	$1.3865\pm 0.0251$	$0.1758\pm 0.0201$	$0.0136\pm 0.0072$

Appendix

Table 10: Matched comparison on SBM (

c{=}16

). aMMD at the validation-converged stage and a fixed nominal NFE budget (Appendix A.6.5 ). Lower is better and bold denotes the best mean in each column. Results are mean

\pm

sample standard deviation over three random seeds.

SETTING	BEST CONTROL	GRCD	RATIO
METR-LA	$0.1921\pm 0.0014$ (full)	$\mathbf{0.0538}$	$\mathbf{3.57\times}$
SBM ( $c{=}16$ )	$0.1232\pm 0.0041$ (spectral)	$\mathbf{0.0421}$	$\mathbf{2.93\times}$
Molene	$0.1822\pm 0.0242$ (spectral)	$\mathbf{0.1114}$	$\mathbf{1.64\times}$
SBM ( $c{=}4$ )	$0.0799\pm 0.0033$ (spectral)	$\mathbf{0.0616}$	$\mathbf{1.30\times}$
SBM ( $c{=}1$ )	$0.0535\pm 0.0027$ (spectral)	$\mathbf{0.0497}$	$\mathbf{1.08\times}$

Appendix

Table 11: Gaussian controls versus GRCD at NFE

4

. aMMD. Lower is better. Gaussian-control values are mean

\pm

sample standard deviation over

5

independent draws. “Best control” denotes the lowest aMMD among the four non-learned Gaussian controls. In the brackets, “full” is the full Ledoit–Wolf Gaussian and “spectral” is the fitted graph-spectral Gaussian reference.

CONFIGURATION	aMMD $\leq 0.2$	aMMD $\leq 0.1$	aMMD $\leq 0.06$
EDM	$4.01$ (NFE 32)	never	never
GAD	never	never	never
WSD-style, scalar	never	never	never
WSD-style, graph prior	$2.07$ (NFE 16)	$8.27$ (NFE 64)	never
Isotropic VE	$1.94$ (NFE 16)	never	never
Static precond.	$1.94$ (NFE 16)	never	never

Appendix

Table 12: Cost to reach a quality target on METR-LA. Cheapest measured sampling cost, in seconds per

1000

samples, at which each configuration first reaches an aMMD target at

\kappa=2

, converged. “never” means the target is not reached at any budget up to NFE

64

. “

+

fitted terminal reference” is the row called “GRCD w/o residual score” in Table 1 .

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