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[[Zhang]] et al. (2024) <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea equates causal emergence with dynamical reversibility. The [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value. This index is highly correlated with EI and can also be used to characterize the causal effect strength of  dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values, without requiring a coarse-graining scheme.

Zhang et al. (2024) <ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> introduced a new causal emergence theory using [[singular value decomposition]].  The core idea equates causal emergence with dynamical reversibility. The [[reversibility measure]] of Markov dynamics (<math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math>) is defined as the sum of <math>\alpha</math> power of its singular values. Here, [math]\sigma_i[/math] represents the singular value. This index is highly correlated with EI and can also be used to characterize the causal effect strength of  dynamics. This method defines '''clear emergence''' and '''vague emergence''' directly from the spectrum of singular values, without requiring a coarse-graining scheme.

== Quantification of causal emergence ==

== Quantification of causal emergence ==