Towards Identifiability of Hierarchical Temporal Causal Representation Learning
Zijian Li, Minghao Fu, Junxian Huang, Yifan Shen, Ruichu Cai, Yuewen Sun, Guangyi Chen, Kun Zhang
TL;DR
This work tackles identifiability of hierarchical temporal causal representations in time series by introducing CHiLD, a theory-guided framework that recovers multi-layer latent dynamics from temporal context. The authors prove identifiability results: (i) ${2L+1}$ adjacent observations suffice to identify the joint multi-layer latents up to invertible transforms, and (ii) within each layer, block-wise identifiability follows from sparsity and conditional independence, leading to component-wise identifiability. The CHiLD model implements these ideas with a contextual encoder and normalizing-flow-based hierarchical priors within a VAE, enabling controllable, hierarchical time-series generation. Empirical results on synthetic and real-world datasets show CHiLD outperforms strong baselines in identifiability, generation quality, and controllability, illustrating practical impact for domains like finance, climate, and biomechanical data.
Abstract
Modeling hierarchical latent dynamics behind time series data is critical for capturing temporal dependencies across multiple levels of abstraction in real-world tasks. However, existing temporal causal representation learning methods fail to capture such dynamics, as they fail to recover the joint distribution of hierarchical latent variables from \textit{single-timestep observed variables}. Interestingly, we find that the joint distribution of hierarchical latent variables can be uniquely determined using three conditionally independent observations. Building on this insight, we propose a Causally Hierarchical Latent Dynamic (CHiLD) identification framework. Our approach first employs temporal contextual observed variables to identify the joint distribution of multi-layer latent variables. Sequentially, we exploit the natural sparsity of the hierarchical structure among latent variables to identify latent variables within each layer. Guided by the theoretical results, we develop a time series generative model grounded in variational inference. This model incorporates a contextual encoder to reconstruct multi-layer latent variables and normalize flow-based hierarchical prior networks to impose the independent noise condition of hierarchical latent dynamics. Empirical evaluations on both synthetic and real-world datasets validate our theoretical claims and demonstrate the effectiveness of CHiLD in modeling hierarchical latent dynamics.