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Jacobian Regularizer-based Neural Granger Causality
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Proposes JRNGC, which learns summary and full-time Granger causality with a single residual MLP forecaster trained with an input-output Jacobian regularizer (L1 or Frobenius norm) instead of per-variable models with first-layer sparsity.
Problem
Existing neural Granger causality methods need one model per target variable and rely on first-layer weight sparsity, which limits accuracy, scalability, and the ability to recover full-time (lagged) causality, especially for RNNs.
Approach
Replace first-layer weight sparsity with a penalty on the input-output Jacobian of a single multivariate forecaster (L1 or squared Frobenius norm, the latter estimated via random projections), and read Granger causality from the Jacobian in post-hoc analysis.
Key results
- On VAR(100,5,10) with lag, JRNGC-L1 reaches AUROC 1.000 and AUPRC 1.000, matching JGC and above cMLP (0.973/0.770).
- On fMRI summary causal graph, JRNGC-F achieves AUPRC 0.749 versus cMLP 0.726, while JGC has the best AUROC (0.912 vs 0.898 for JRNGC-F).
- On CausalTime, JRNGC-F reaches AUROC 0.9279 on AQI and 0.7294 on Traffic, the reported best on those datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the input-output Jacobian regularizer (ablation) | JRNGC-F on VAR100 · VAR100 summary | AUPRC: 0.667 -> 0.950 | Helped |
| Add the input-output Jacobian regularizer (ablation) | JRNGC-F on fMRI · fMRI summary | AUPRC: 0.266 -> 0.749 | Helped |
| Add the input-output Jacobian regularizer (ablation) | JRNGC-F on Lorenz-96 · Lorenz-96 summary | AUROC: 0.813 -> 0.875 | Helped |
| Add 1 residual layer (ablation) | JRNGC-F · VAR100 summary | AUPRC: 0.950 -> 0.894 | Hurt |
| Add 1 residual layer (ablation) | JRNGC-F · Lorenz-96 summary | AUPRC: 0.692 -> 0.959 | Helped |
| Replace F-norm Jacobian regularizer with DAG regularizer (ablation) | JRNGC on VAR(100, 5, 10) · VAR (100, 5, 10) | AUPRC: 0.959 -> 0.922 | Hurt |
| Replace F-norm Jacobian regularizer with DAG regularizer (ablation) | JRNGC on DREAM4 · DREAM4 | AUPRC: 0.253 -> 0.042 | Hurt |
| Add Jacobian regularizer (ablation) | LSTM on Lorenz-96 F=40 · Lorenz-96 F=40, full-time | AUPRC: $0.538 \pm 0.045$ -> $\textbf{0.929} \pm 0.014$ | Helped |
| Add Jacobian regularizer (ablation) | LSTM on Lorenz-96 F=40 · Lorenz-96 F=40, summary | AUROC: $0.661 \pm 0.040$ -> $\textbf{0.942} \pm 0.019$ | Helped |
| Remove the Jacobian regularizer (ablation) | JRNGC-F on Lorenz-96 F=40 · Lorenz-96 F=40 | MSE: 0.878 -> 1.269 | Hurt |
| Use JRNGC-F instead of the strongest displayed baseline, CUTS+ | CausalTime · CausalTime AQI | AUROC: 0.8928 -> 0.9279 | Helped |
| Use JRNGC-F instead of the strongest displayed baseline, CUTS+ | CausalTime · CausalTime Traffic | AUROC: 0.6329 -> 0.7294 | Helped |
Surprising findings
- The DAG regularizer on the Jacobian performed worse than the sparsity regularizer on the DREAM4 dataset with cyclic ground truth (AUROC 0.614 vs 0.766).
Limitations
- Performance on DREAM-3 is weak overall due to very few observations, causing overfitting.
- JRNGC-L1 training on VAR100 took about 9.8 s per epoch, far slower than JRNGC-F, indicating L1 Jacobian penalty cost.
- Significance thresholds rely on surrogate datasets and a 2-sigma rule, which is a heuristic.
Experimental setup
Datasets: VAR synthetic Lorenz-96 fMRI (Smith et al. simulations) DREAM-3 DREAM-4 CausalTime epilepsy EEG
Benchmarks: VAR Lorenz-96 fMRI DREAM-3 DREAM-4 CausalTime
Metrics: AUROC AUPRC
Baselines: GC PCMCI cMLP cLSTM NAVAR (MLP) NAVAR (LSTM) JGC CR-VAE eSRU TCDF CUTS+ LCCM
Builds on: cMLP cLSTM NAVAR JGC Jacobian regularization
Closest papers by meaning
- Exploring Neural Granger Causality with xLSTMs: Unveiling Temporal Dependencies in Complex Data (NeurIPS 2025 · similarity 0.90)
- TNPAR: Topological Neural Poisson Auto-Regressive Model for Learning Granger Causal Structure from Event Sequences (AAAI 2024 · similarity 0.87)
- Towards the Disappearing Truth: Fine-Grained Joint Causal Influences Learning with Hidden Variable-Driven Causal Hypergraphs in Time Series (AAAI 2024 · similarity 0.87)
- Federated Granger Causality Learning For Interdependent Clients With State Space Representation (ICLR 2025 · similarity 0.85)
- CUTS+: High-Dimensional Causal Discovery from Irregular Time-Series (AAAI 2024 · similarity 0.85)
- GCAD: Anomaly Detection in Multivariate Time Series from the Perspective of Granger Causality (AAAI 2025 · similarity 0.85)
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