TopoDistill: Distilling Global System Topology for Causal Discovery in Multivariate Time Series

Although causal discovery from multivariate time series is widely used, it remains challenging under noise. Convergent cross mapping (CCM) infers causality by reconstructing shadow manifolds via time-delay embedding (TDE) and evaluating cross-map skill between manifolds. Despite Takens’ theorem guarantees in ideal settings, TDE effectively attempts to recover system state from a single noisy view, often yielding geometrically degraded manifolds and unreliable distance-based neighborhoods, which in turn weakens causal identification. We propose TopoDistill, a topology-informed knowledge distillation framework that improves univariate shadow-manifold reconstruction by aligning local neighborhood structure to a multivariate system representation. A global embedder trained on multivariate observations captures a global attractor representation, while a delay embedder is distilled to produce embeddings whose neighborhood distributions match the global topology. This cross-view alignment yields smoother and more reliable neighborhoods, improving cross mapping under noise while maintaining specificity against spurious correlations. Theoretical analysis and experimental results demonstrate that our method enables effective causal discovery.

Understanding cause-and-effect in complex systems—like the human body or the Earth's climate—is crucial, but real-world data is inherently noisy and confusing. Traditional methods try to find these connections by recreating the system's underlying "shape" from just a single stream of data. However, measurement noise easily distorts this picture, leading to unreliable conclusions. To solve this, we developed TopoDistill. Think of it like a teacher guiding a student. First, we look at all available data streams together to get a clear, "global map" of how the system actually works. Then, we use this accurate map as a template to teach our model how to perfectly reconstruct the system using only a single, noisy data stream. This alignment smooths out the noise and preserves the true underlying structure. Ultimately, TopoDistill allows us to confidently identify genuine cause-and-effect relationships hidden in chaotic, real-world data, providing a far more reliable tool for scientific discovery.