Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts

2026-06-26

AI Summary

This work addresses the challenge of non-identifiability in existing continuous-time latent-variable stochastic differential equation (SDE) models under nonlinear observations. The authors propose a novel approach based on additive-noise latent SDEs with shared drift but environment-dependent diffusion covariances, demonstrating that only two diagonal diffusion mechanisms suffice to disentangle latent variables via coordinate-wise variance ratio analysis—without requiring sparsity assumptions on the drift term. Theoretically, they establish, for the first time, identifiability of the latent variables up to permutation and scaling, and further recover the instantaneous causal graph encoded by the drift’s Jacobian. Leveraging a diffusivity-shift-based identifiability framework and a two-stage algorithm, the method successfully validates theoretical identifiability bounds on synthetic data and effectively disentangles real-world sensor trajectories from the Hardanger Bridge, demonstrating practical efficacy.

Abstract

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

Problem

• identifiability
• latent SDEs
• continuous-time
• causal representation learning
• diffusion covariance

Innovation

• latent SDEs
• identifiability
• diffusion shifts
• causal representation learning
• continuous-time dynamics