  Identifying the underlying time-delayed latent causal processes in sequential
data is vital for grasping temporal dynamics and making downstream reasoning.
While some recent methods can robustly identify these latent causal variables,
they rely on strict assumptions about the invertible generation process from
latent variables to observed data. However, these assumptions are often hard to
satisfy in real-world applications containing information loss. For instance,
the visual perception process translates a 3D space into 2D images, or the
phenomenon of persistence of vision incorporates historical data into current
perceptions. To address this challenge, we establish an identifiability theory
that allows for the recovery of independent latent components even when they
come from a nonlinear and non-invertible mix. Using this theory as a
foundation, we propose a principled approach, CaRiNG, to learn the CAusal
RepresentatIon of Non-invertible Generative temporal data with identifiability
guarantees. Specifically, we utilize temporal context to recover lost latent
information and apply the conditions in our theory to guide the training
process. Through experiments conducted on synthetic datasets, we validate that
our CaRiNG method reliably identifies the causal process, even when the
generation process is non-invertible. Moreover, we demonstrate that our
approach considerably improves temporal understanding and reasoning in
practical applications.
