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

Area of Science:

• Complex Systems
• Information Theory
• Dynamical Systems Theory

Background:

• Causal emergence (CE) highlights how macrostates can possess greater causal influence than microstates.
• Identifying CE and maximizing effective information (EI) traditionally requires challenging coarse-graining strategies.
• Existing CE frameworks are often limited to discrete states or specific dynamical properties.

Purpose of the Study:

• To propose a novel CE quantification framework for Gaussian iterative systems.
• To develop a method independent of coarse-graining, utilizing approximate dynamical reversibility.
• To establish precise coarse-graining strategies derived from singular value decomposition (SVD) spectra.

Main Methods:

• Employing SVD on inverse covariance matrices for forward and backward dynamics to assess approximate dynamical reversibility.
• Analytically deriving the positive correlation and equivalence condition between SVD-based and EI-based CE.
• Developing coarse-graining strategies directly from singular value spectra and orthogonal matrices.

Main Results:

• A new CE quantification framework for Gaussian systems with continuous states and noise is presented.
• The framework demonstrates a positive correlation with EI-based CE and provides an equivalence condition.
• Precise coarse-graining strategies are derived from SVD spectra, applicable to various dynamical systems.

Conclusions:

• The proposed framework offers a robust method for quantifying CE in Gaussian iterative systems, independent of coarse-graining.
• It extends CE analysis to continuous systems and highlights the role of noise and covariance.
• This approach has broad applications, including autoregressive models, Markov-Gaussian systems, and machine learning models like SIR networks.