SC3D: Dynamic and Differentiable Causal Discovery for Temporal and Instantaneous Graphs

Discovering causal structures from multivariate time series is a key problem because interactions span across multiple lags and possibly involve instantaneous dependencies. Additionally, the search space of the dynamic graphs is combinatorial in nature. In this study, we propose \textit{Stable Causal Dynamic Differentiable Discovery (SC3D)}, a two-stage differentiable framework that jointly learns lag-specific adjacency matrices and, if present, an instantaneous directed acyclic graph (DAG). In Stage 1, SC3D performs edge preselection through node-wise prediction to obtain masks for lagged and instantaneous edges, whereas Stage 2 refines these masks by optimizing a likelihood with sparsity along with enforcing acyclicity on the instantaneous block. Numerical results across synthetic and benchmark dynamical systems demonstrate that SC3D achieves improved stability and more accurate recovery of both lagged and instantaneous causal structures compared to existing temporal baselines.

💡 Research Summary

The paper introduces SC3D (Stable Causal Dynamic Differentiable Discovery), a novel two‑stage differentiable framework for learning both lag‑specific and instantaneous causal relationships from multivariate time‑series data. The authors model the data as a structural vector autoregressive (SVAR) process: (X_{t+1}=B^{\star}X_t+\sum_{\ell=1}^{L}A_{\ell}^{\star}X_{t+1-\ell}+\varepsilon_{t+1}), where each (A_{\ell}^{\star}) captures directed effects across (\ell) time steps and (B^{\star}) encodes instantaneous intra‑slice dependencies. Crucially, (B^{\star}) must form a directed acyclic graph (DAG) to guarantee a well‑defined structural model, while lagged edges are inherently acyclic due to temporal ordering.

Stage 1 – Node‑wise Temporal Preselection
For each target variable (X_j^{t+1}) a candidate parent window (V^{(j)}_t) is constructed, containing all lagged variables up to order (L) and, if allowed, contemporaneous variables (excluding self‑loops). A differentiable predictor (e.g., a neural network with grouped input weights) is trained to maximize a penalized conditional log‑likelihood:
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