Extend CE quantification beyond locally linear approximation for nonlinear GIS

Develop a causal-emergence quantification framework for nonlinear Gaussian iterative systems x_{t+1}=f(x_t)+ε_t that does not rely solely on the locally linear approximation A(x)=∇f(x), remains stable when the Jacobian ∇f(x) is zero or unbounded, and robustly defines reversible information (e.g., via higher-order derivatives) so that the framework does not break down due to ill-conditioned gradients.

Background

The paper develops an SVD-based causal emergence (CE) framework for linear Gaussian iterative systems (GIS) and extends it to nonlinear settings by locally linearizing the dynamics using the Jacobian, A(x)=∇f(x).

However, the authors note that this linearization can be unstable: when the gradient vanishes or becomes unbounded, the method can fail. They suggest that incorporating higher-order derivatives could address this limitation, indicating a need for a more general and robust CE formulation for nonlinear GIS.