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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2210.03850 |
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| _version_ | 1866912644821155840 |
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| author | Li, Xin |
| author_facet | Li, Xin |
| contents | We introduce \emph{Information Topology}: a framework that unifies information theory and algebraic topology by treating \emph{cycle closure} as the primitive operation of inference. The starting point is the \emph{dot-cycle dichotomy}, which separates pointwise, order-sensitive fluctuations (dots) from order-invariant, predictive structure (cycles). Algebraically, closure is the cancellation of boundaries ($\partial^2=0$), which converts transient histories into stable invariants. Building on this, we derive the \emph{Structure-Before-Specificity} (SbS) principle: stable information resides in nontrivial homology classes that persist under perturbations, while high-entropy contextual details act as scaffolds. The \emph{Context-Content Uncertainty Principle} (CCUP) quantifies this balance by decomposing uncertainty into contextual spread and content precision, showing why prediction requires invariance for generalization. Measure concentration onto residual invariant manifolds explains \emph{order invariance}: when mass collapses to a narrow tube around a closed cycle, reparameterizations of micro-steps leave predictive functionals unchanged. We then define \emph{homological capacity}, the topological dual of Shannon capacity, as the sustainable number of independent informational cycles supported by a system. This capacity links dynamical (KS) entropy to structural (homological) capacity and refines Euler characteristics from a ``net'' summary to a ``gross'' count of persistent invariants. Finally, we illustrate the theory across three domains where \emph{more is different}: \textbf{visual binding}, \textbf{working memory}, and \textbf{access consciousness}. Together, these results recast inference, learning, and communication as \emph{topological stabilization}: the formation, closure, and persistence of informational cycles that make prediction robust and scalable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03850 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Information Topology Li, Xin Computer Vision and Pattern Recognition We introduce \emph{Information Topology}: a framework that unifies information theory and algebraic topology by treating \emph{cycle closure} as the primitive operation of inference. The starting point is the \emph{dot-cycle dichotomy}, which separates pointwise, order-sensitive fluctuations (dots) from order-invariant, predictive structure (cycles). Algebraically, closure is the cancellation of boundaries ($\partial^2=0$), which converts transient histories into stable invariants. Building on this, we derive the \emph{Structure-Before-Specificity} (SbS) principle: stable information resides in nontrivial homology classes that persist under perturbations, while high-entropy contextual details act as scaffolds. The \emph{Context-Content Uncertainty Principle} (CCUP) quantifies this balance by decomposing uncertainty into contextual spread and content precision, showing why prediction requires invariance for generalization. Measure concentration onto residual invariant manifolds explains \emph{order invariance}: when mass collapses to a narrow tube around a closed cycle, reparameterizations of micro-steps leave predictive functionals unchanged. We then define \emph{homological capacity}, the topological dual of Shannon capacity, as the sustainable number of independent informational cycles supported by a system. This capacity links dynamical (KS) entropy to structural (homological) capacity and refines Euler characteristics from a ``net'' summary to a ``gross'' count of persistent invariants. Finally, we illustrate the theory across three domains where \emph{more is different}: \textbf{visual binding}, \textbf{working memory}, and \textbf{access consciousness}. Together, these results recast inference, learning, and communication as \emph{topological stabilization}: the formation, closure, and persistence of informational cycles that make prediction robust and scalable. |
| title | Information Topology |
| topic | Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2210.03850 |