Causal Representation Meets Stochastic Modeling under Generic Geometry

We investigate the identifiability problem of latent stochastic processes characterized by high dynamics that occur in continuous time with varying intensities (e.g., a multivariate Hawkes process), and we provide the corresponding identifiability theory. Building on this theoretical foundation, we implement MUTATE, a variational autoencoder framework with a time-adaptive transition module to evaluate stochastic dynamics on both synthetic stochastic processes and real-world biological signal data. This work advances causal representation learning theory by extending it to continuous-time and stochastic settings via weak topology and algebraic signature, highlighting the importance of this approach in addressing scientific questions, such as the accumulation of mutations in genomics and the mechanisms driving neuron spike triggers in response to time-varying dynamics.