{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/dual-valued-functions-of-dual-matrices-with","title":"Dual-Valued Functions of Dual Matrices with Applications in Causal Emergence","arxiv_id":"2411.08377","date":"2024-11-13","proceeding":null,"authors":["Tong Wei","Weiyang Ding","Yimin Wei"],"abstract":"Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the G\\^ateaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector $p$-norm $(1\\!\\leq\\! p\\!\\leq\\!\\infty)$ and the unitarily invariant dual-valued Ky Fan $p$-$k$-norm $(1\\!\\leq\\! p\\!\\leq\\!\\infty)$. The equivalence between the dual-valued Ky Fan $p$-$k$-norm and the dual-valued vector $p$-norm of the first $k$ singular values of the dual matrix is then demonstrated. Practically, we define the dual transitional probability matrix (DTPM), as well as its dual-valued effective information (${\\rm{EI_d}}$). Additionally, we elucidate the correlation between the ${\\rm{EI_d}}$, the dual-valued Schatten $p$-norm, and the dynamical reversibility of a DTPM. Through numerical experiments on a dumbbell Markov chain, our findings indicate that the value of $k$, corresponding to the maximum value of the infinitesimal part of the dual-valued Ky Fan $p$-$k$-norm by adjusting $p$ in the interval $[1,2)$, characterizes the optimal classification number of the system for the occurrence of the causal emergence.","url_abs":"https://arxiv.org/abs/2411.08377v1","url_pdf":"https://arxiv.org/pdf/2411.08377v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"dual-valued-functions-of-dual-matrices-with","repo_url":"https://github.com/TongWeiviolet/Causal-emergence-analysis","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}