Information-Theoretic Obstructions to Embedding Reconstruction: Sharp Constants via Alpay Operator Theory and Categorical Frameworks

Information-Theoretic Obstructions to Embedding Reconstruction: Sharp Constants via Alpay Operator Theory and Categorical Frameworks | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 9 June 2025 V1 Latest version Share on Information-Theoretic Obstructions to Embedding Reconstruction: Sharp Constants via Alpay Operator Theory and Categorical Frameworks Author : Faruk Alpay 0009-0009-2207-6528 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.174948658.89709607/v1 177 views 114 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This article establishes sharp information-theoretic limits on reconstructing high-dimensional embeddings from probabilistic outputs. For a system with embedding dimension d and output entropy H, any reconstruction ê satisfies E[∥ê − e∥ 2 2 ] ≥ 1 2πe • d H 2, with equality achieved for Gaussian distributions. The proof employs the Alpay ⋆-algebra structure, whose derivation property yields fixed-point theorems via Grothendieck's adjoint functor theorem. It is proved that the Alpay entropic operator Φ creates reconstruction barriers when ∥Φ∥ op > e Hc. Spectral formulas λ 1 = 1 − exp(−H) connecting to optimal transport are derived, and a categorical framework is developed where reconstruction morphisms factor through the initial object φ ∞. The constant 1 2πe emerges from Shannon's entropy power inequality through spectral analysis. Algorithms achieving these bounds are provided along with topological invariants. Permanently archived on Arweave. Supplementary Material File (information_theoretic_obstructions_to_embedding_reconstruction__sharp_constants_via_alpay_operator_theory_and_categorical_frameworks.pdf) Download 558.30 KB Information & Authors Information Version history V1 Version 1 09 June 2025 Copyright This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License Keywords alpay algebra approximation theory category theory computational geometry convex optimization entropic bounds functional analysis generative models high-dimensional embeddings information theory manifold learning metric geometry non-commutative geometry operator algebras optimal transport quantum information theory reconstruction limits representation learning sheaf theory statistical physics topological data analysis Authors Affiliations Faruk Alpay 0009-0009-2207-6528 [email protected] Independent Researcher Adana View all articles by this author Metrics & Citations Metrics Article Usage 177 views 114 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Faruk Alpay. Information-Theoretic Obstructions to Embedding Reconstruction: Sharp Constants via Alpay Operator Theory and Categorical Frameworks. Authorea . 09 June 2025. DOI: https://doi.org/10.22541/au.174948658.89709607/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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