Go to TMLR homepageCauFR-TS: Causal Time-Series Identifiability via Factorized Representations
Abstract: Causal discovery from multivariate time series is a fundamental problem for interpretable modelling, causality-aware downstream analysis, and intervention-driven simulation. Recent neural approaches commonly rely on shared latent embeddings to capture temporal dynamics and utilize them for causal structure estimation and downstream prediction. We formally establish that such shared encoders entangle distinct causal mechanisms into a unified latent manifold, which exhibits fundamental theoretical limitations of structural non-identifiability and conditional independence assumptions required for Granger causality. To address these issues, we propose CauFR-TS, a recurrent variational framework that enforces mechanism modularity through dimension-wise encoders and ensures mediation of all cross-variable dependencies through structured latent aggregation. Furthermore, we address the instability of heuristic thresholding in continuous relaxation methods by proposing an adaptive, data-driven unsupervised link selection strategy based on decoder weight distribution. Empirical evaluation on synthetic and in silico biological benchmarks demonstrates that CauFR-TS outperforms recent baselines in graph recovery metrics while preserving competitive probabilistic forecasting performance.
Submission Type: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: This revision addresses both points raised by the Action Editor.
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**1. Strengthened theoretical analysis (Appendix D, Pages 19–22).** The previous proofs of Propositions 1 and 2 contained two informal steps: an optimizer-behaviour argument for Proposition 1 and an approximate single-neurone formulation for Proposition 2. Both of them have been replaced with formal derivations. We introduce **Lemma 1 (Generic Density of the Shared Encoder Jacobian, Page 19)**, which establishes that every entry of the encoder Jacobian is non-zero for Lebesgue-almost every parameter–input pair. The proof rests on (i) generic invertibility of activation derivative matrices under Assumption (A1), and (ii) a polynomial non-vanishing argument via [1] showing that each Jacobian entry, being a non-identically-zero polynomial in the parameters, vanishes only on a measure-zero set. This lemma serves as the common formal foundation for both propositions. The proof of **Proposition 1 (Section D.1, Page 20)** now invokes Lemma 1 directly: the generically non-zero off-diagonal Jacobian blocks contradict the block-diagonal structure required for variable-wise separability. The proof of **Proposition 2 (Section D.2, Pages 20–21)** is restructured into four explicit steps using an exact inner-product formulation $\frac{\partial \hat{x}^{(i)}\_{t}}{\partial \hat{x}^{(j)}\_{t-}} = \mathbf{d}_i^\top \mathbf{e}_j$, where $\mathbf{d}\_i$ and $\mathbf{e}\_j$ denote the decoder and encoder sensitivity vectors. The proof shows that this inner product is a non-identically-zero polynomial via a concrete witness configuration and concludes via the same measure-theoretic argument as Lemma 1. The remark on the identity map (Section D.2, Page 21) and the justification of Remark 1 (Section D.3, Pages 21–22) are retained with updated back-references to Lemma 1.
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**2. Expanded empirical evaluation (Section 5.1, Tables 1–2, Appendix A and F).** The original four-dataset evaluation did not adequately probe the diversity of graph topologies and dynamical regimes. We have added two synthetic benchmarks, broadening the evaluation along complementary axes of graph topology and nonlinearity structure, and evaluated under the same protocol with mean $\pm$ standard deviation across 5 random seeds.
- **CF-Diamond (Section 5.1, Page 8; Appendix A, Pages 16–17).** A four-variable diamond benchmark adopted from [2], combining mediators, a collider, and self-causal links with mixed lagged and instantaneous dependencies. It complements the chain topology of Hénon and the dense topology of Lorenz-96 by probing recovery under overlapping ancestral paths, a canonical failure mode for shared-encoder architectures. Adopting it from a strong baseline (CausalFormer) ensures the comparison favours that baseline.
- **NL-VAR (Section 5.1, Page 8; Appendix A, Page 17).** A 10-dimensional first-order autoregressive process with quadratic parent interactions and a smooth saturating transition on a random sparse topology with heterogeneous per-edge coupling, probing recovery under stochastic saturating nonlinear regimes with non-uniform effect sizes.
The six benchmarks now span chaotic, dissipative, and saturating nonlinear dynamics; deterministic and stochastic processes; chain, dense, diamond, and random topologies; and synthetic and biologically grounded networks. Quantitative results appear in Tables 1 and 2 (Page 11); convergence, qualitative recovery, posterior density, and t-SNE analyses on the new benchmarks are provided in Figures 6–9 (Pages 27–30); expanded discussion is in Section 6.1 (Page 10). On NL-VAR, CauFR-TS attains the highest AUROC (0.944), highest F1 (0.853), and lowest SHD (12). On CF-Diamond, it achieves perfect ranking (AUROC = 1.0) but a marginally lower F1 (0.659) than CausalFormer and TCDF (0.681); the gap is attributable to the small-graph regime, where only 16 weight norms are available to the GMM, while the perfect AUROC confirms correct edge ranking. CauFR-TS also attains the lowest one-step-ahead RMSE on every benchmark, including the two new systems (Table 2, Page 11).
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**References**
[1] J. T. Schwartz. 1980. Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. ACM 27, 4 (Oct. 1980), 701–717. https://doi.org/10.1145/322217.322225
[2] Lingbai Kong, Wengen Li, Hanchen Yang, Yichao Zhang, Jihong Guan, and Shuigeng Zhou. Causalformer: An interpretable transformer for temporal causal discovery. IEEE Transactions on Knowledge and Data Engineering, 37(1):102–115, 2025. doi: 10.1109/TKDE.2024.3484461.
Code: https://github.com/ayan-cs/caufr-ts
Assigned Action Editor: ~Hongfu_Liu2
Submission Number: 7403
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