Singular-value-decomposition-based causal emergence for Gaussian iterative systems

Causal emergence (CE) based on effective information (EI) demonstrates that macrostates can exhibit stronger causal effects than microstates in dynamics. However, the identification of CE and the maximization of EI both rely on coarse-graining strategies, which is a key challenge. A recently proposed CE framework based on approximate dynamical reversibility, utilizing singular-value decomposition (SVD), is independent of coarse-graining. Still, it is limited to transition probability matrices in discrete states. To address this, this article proposes a CE quantification framework for Gaussian iterative systems, based on approximate dynamical reversibility derived from the SVD of inverse covariance matrices in forward and backward dynamics. The positive correlation between SVD-based and EI-based CE, along with the equivalence condition, is given analytically. After that, we provide precise coarse-graining strategies directly from singular-value spectra and orthogonal matrices. This new framework can be applied to any dynamical system with continuous states and Gaussian noise, such as autoregressive growth models, Markov-Gaussian systems, and even SIR modeling using neural networks. Numerical simulations on typical cases validate our theory and offer a new approach to studying the CE phenomenon, emphasizing noise and covariance over dynamical functions in both known models and machine learning.