Smoothing the Landscape: Causal Structure Learning via Diffusion Denoising Objectives

Hao Zhu; Di Zhou; Donna Slonim

cs.LGApr 2, 2026

Smoothing the Landscape: Causal Structure Learning via Diffusion Denoising Objectives

Authors: Hao Zhu, Di Zhou, Donna Slonim

Organizations: Beth Israel Deaconess Medical Center Harvard Medical School Boston, MA, USA · Department of Computer Science Tufts University Medford, MA, USA

Abstract

Understanding causal dependencies in observational data is critical for informing decision-making. These relationships are often modeled as Bayesian Networks (BNs) and Directed Acyclic Graphs (DAGs). Existing methods, such as NOTEARS and DAG-GNN, often face issues with scalability and stability in high-dimensional data, especially when there is a feature-sample imbalance. Here, we show that the denoising score matching objective of diffusion models could smooth the gradients for faster, more stable convergence. We also propose an adaptive k-hop acyclicity constraint that improves runtime over existing solutions that require matrix inversion. We name this framework Denoising Diffusion Causal Discovery (DDCD). Unlike generative diffusion models, DDCD utilizes the reverse denoising process to infer a parameterized causal structure rather than to generate data. We demonstrate the competitive performance of DDCDs on synthetic benchmarking data. We also show that our methods are practically useful by conducting qualitative analyses on two real-world examples. Code is available at this url: https://github.com/haozhu233/ddcd.

Figures & tables

Figure 1: Model architectures of proposed models in this paper. Note that the reverse path does not generate new samples and the only focus is to learn W via the denoising objective.

Figure 2: The denoising objective accelerates convergence by smoothing gradients. (a) Average runtime vs. sample size (10 runs). (b) L-BFGS-B steps to convergence; NOTEARS-Denoising requires significantly fewer iterations. (c) Gradient norm trajectories during the first iteration on full data ( $n=2,000$ ); the denoising objective yields a stable descent compared to the volatile gradients of the linear baseline. (First 5 steps omitted for clarity).

Figure 3: DDCD Linear and Nonlinear models demonstrate robust scalability and accuracy on synthetic benchmarks.

Figure 4: Inferred Causal Network around Lethal Outcome in Myocardial Infarction

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Example Weight Estimates on Graphs with 1,000 nodes. Number of samples in both cases is 2,000.

Figure 6: Comparison of some results from NOTEARS-Linear and NOTEARS-Denoising. Execution times are displayed on the top-right corner of each figure.

Optimizer	Nodes	SHD	TPR	FPR	FDR	Time(s)
Augmented Lagrangian	20	32.5±3.0	0.45±0.03	0.01±0.01	0.04±0.02	15.8±8.2
	50	135.8±11.7	0.23±0.04	0.00±0.00	0.10±0.08	13.8±0.8
	100	328.5±18.1	0.13±0.01	0.00±0.00	0.15±0.03	27.4±8.1
Linear Multiplier	20	10.7±5.9	0.96±0.04	0.07±0.04	0.14±0.07	8.9±1.7
	50	46.2±12.5	0.97±0.01	0.04±0.01	0.21±0.05	12.8±2.1
	100	139.3±16.2	0.96±0.01	0.03±0.00	0.26±0.03	15.9±3.3

Appendix

Table 1: Optimization Strategy Comparison: Augmented Lagrangian vs. Linear Multiplier

Figure 7: Change of DAG constraint h under different k-hop schedules

Figure 8: Effect of Diffusion timestep and noise schedule

N	Data	Graph	Method	SHD	TPR	FDR	FPR	Time(s)
2000	linear	SF-20	DAG-GNN	0.9±1.0	0.99±0.01	0.02±0.02	0.01±0.01	548.8
2000	linear	SF-20	DDCD Linear	0.3±0.9	1.00±0.01	0.01±0.02	0.00±0.01	13.1
2000	linear	SF-20	DDCD Non-linear	9.0±18.3	0.87±0.25	0.07±0.19	0.02±0.04	4.1
2000	linear	SF-20	DDCD Smooth	50.7±7.0	0.17±0.16	0.73±0.18	0.17±0.06	4.0
2000	linear	SF-20	GAE	139.3±3.2	0.60±0.03	0.86±0.00	1.43±0.10	890.5
2000	linear	SF-20	GOLEM	6.0±0.0	0.92±0.00	0.08±0.00	0.03±0.00	156.4

Appendix

Table 2: Main benchmark results (SHD, TPR, FDR, FPR across all methods)

Figure 10: SHD Results from the Non-linear Diversity Experiments

Figure 11: Time Costs from the Non-linear Diversity Experiments

Figure 12: Inferred Causal Network in an Aging Study.

Source	Source Label	Target	Target Label	Weight
LID_KB	Use of lidocaine by the Emergency Cardiology Team	NA_KB	Use of opioid drugs by the Emergency Cardiology Team	0.66
R_AB_2_n	Relapse of the pain in the second day of the hospital period	NA_R_2_n	Use of opioid drugs in the ICU in the second day of the hospital period	0.58
R_AB_1_n	Relapse of the pain in the first hours of the hospital period	NA_R_1_n	Use of opioid drugs in the ICU in the first hours of the hospital period	0.52
R_AB_3_n	Relapse of the pain in the third day of the hospital period	NA_R_3_n	Use of opioid drugs in the ICU in the third day of the hospital period	0.50
NOT_NA_2_n	Use of NSAIDs in the ICU in the second day of the hospital period	R_AB_2_n	Relapse of the pain in the second day of the hospital period	0.49
RAZRIV	Myocardial rupture	LET_IS	Lethal outcome	0.48

Appendix

Table 3: DDCD Smooth MI Edges

Figure 13: Inferred Causal Network around Lethal Outcome in Myocardial Infarction by NOTEARS.

Source	Source Label	Target	Target Label	Weight
LET_IS	Lethal outcome	AGE	Age of patient.	1
LET_IS	Lethal outcome	S_AD_ORIT	Systolic blood pressure according to intensive care unit	1
LET_IS	Lethal outcome	D_AD_ORIT	Diastolic blood pressure according to intensive care unit	1
FK_STENOK	Functional class (FC) of angina pectoris in the last year.	NA_BLOOD	Serum sodium content	1
MP_TP_POST	Paroxysms of atrial fibrillation at the time of admission to intensive care unit, (or at a pre-hospital stage)	NA_BLOOD	Serum sodium content	1
nr_11	Observing of arrhythmia in the anamnesis	NA_BLOOD	Serum sodium content	1

Appendix

Table 4: NOTEARS MI Edges

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Causal discovery from observational data remains challenging due to the need to recover directed structure and latent confounding without interventions. We propose FoundCause, an amortized causal discovery model trained entirely on synthetic data that maps datasets directly to causal graphs in a single forward pass. By learning from large collections of simulated structural causal models, FoundCause captures transferable statistical patterns that generalize beyond individual datasets. The architecture incorporates several key inductive biases for causal discovery. It uses a permutation-invariant transformer encoder with alternating attention over samples and variables to jointly model cross-variable dependence and per-variable distributions. Pairwise statistical features derived from classical asymmetry measures are injected through statistics-conditioned attention, guiding the model toward known causal signals. A factorized decoder separates edge existence from direction, while a triangular refinement module enables reasoning over higher-order causal motifs such as chains and colliders. In addition, a dedicated confounder module based on learnable latent tokens explicitly models hidden common causes, and the model explicitly handles missing data via its masked input representation. To our knowledge, FoundCause is the first amortized causal discovery approach to explicitly model latent confounding. FoundCause outperforms 11 classical non-amortized methods (e.g., PC, GES, NOTEARS-style optimization) and 4 amortized causal discovery methods on 15 real-world datasets, achieving +9.6% improvement in

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Abstract

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