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

TL;DR

SC3D presents a stable, differentiable framework for discovering both lagged and instantaneous causal relations in multivariate time series by combining a Stage 1 node-wise temporal preselection with a Stage 2 constrained refinement that enforces acyclicity on the instantaneous block via spectral penalties. The method jointly estimates lag-specific adjacency matrices $ig\\{A_ o ext{l}\big\race}$ and an instantaneous DAG $B$, ensuring acyclicity only on $B$ to maintain numerical stability. Empirical results on synthetic and benchmark dynamical systems show SC3D achieves improved stability and more accurate recovery of both lagged and instantaneous structures compared to temporal baselines, with strong lagged ranking and meaningful instantaneous structure discovery. The approach scales to higher dimensions and longer lags, and its ablations confirm the necessity of Stage 1 preselection and the 2-cycle penalty for robust instantaneous recovery. The framework promises robust causal discovery in SVAR-like models and suggests potential extensions to nonstationary and online contexts.

Abstract

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.

Figures (5)

• Figure 1: Structural vector autoregressive (SVAR) model with lagged and instantaneous causal dependencies. Nodes represent variables across time slices. Blue arrows denote lagged causal effects encoded by matrices $\{A_\ell\}_{\ell=1}^L$, while red arrows denote instantaneous causal relationships encoded by a directed acyclic graph $B$. SC3D jointly estimates both components, enforcing acyclicity only on the instantaneous block.
• Figure 2: Scalability of causal structure recovery under increasing dimension ($L=3$, $T=200$). Left: Total structural Hamming distance $\mathrm{SHD}_{\mathrm{total}}$, accounting for both lagged and instantaneous errors, shown on a logarithmic scale. Right: Lagged-only structural error $\mathrm{SHD}_A$ shown on a linear scale for methods which model lagged dependencies, enabling a direct comparison for temporal structure recovery.
• Figure 3: Decomposition of structural error into lagged and instantaneous components for the $d$-sweep experiment ($L=3$, $T=200$). Left: lagged structure error. Right: instantaneous structure error.
• Figure 4: Sensitivity of SC3D to lag order $L$ ($d=8$, $T=200$). The total structural error increases gradually as $L$ grows, while the instantaneous component remains well controlled, indicating stable optimization and robustness to increasing temporal depths.
• Figure 5: TVSEM regime switching direction tracking using windowed evaluation (lag order $L=1$). Left: SC3D. Right: baselines (DYNOTEARS and PCMCI+). SC3D displays smoother and temporally consistent score trajectories that fit well to regime changes while baselines show sharper but less stable transitions throughout windows.

Theorems & Definitions (4)

• Theorem 3.1
• Proposition 3.2
• proof
• proof