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2025
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| Accés en línia: | https://doi.org/10.5281/zenodo.17455111 |
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| _version_ | 1866901603504619520 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the geometric structure of high-dimensional vector spaces through the lens of information theory and entropy. We investigate how fundamental concepts such as Shannon entropy, typical sets, and the concentration of measure phenomenon dictate the effective geometry experienced by data distributions. By applying principles from information geometry, we characterize the space of high-dimensional probability distributions as a statistical manifold, where distances and curvatures are defined by information-theoretic metrics like the Fisher information metric. The study reveals that in high dimensions, probability mass concentrates in a thin shell, a counter-intuitive result that has profound implications for statistical inference, machine learning, and data analysis. We demonstrate that the apparent volume of these spaces is governed by entropic quantities, leading to a geometric framework where proximity and structure are naturally expressed in terms of information divergence rather than traditional Euclidean distance. This entropic perspective provides a powerful toolkit for understanding the behavior of algorithms and the intrinsic dimensionality of data in high-dimensional settings. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17455111 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Entropic Geometry of High-Dimensional Vector Spaces SÉRGIO DE ANDRADE, PAULO High-Dimensional Geometry Information Geometry Entropy Concentration of Measure Statistical Manifolds Typical Sets Fisher Information Metric This paper explores the geometric structure of high-dimensional vector spaces through the lens of information theory and entropy. We investigate how fundamental concepts such as Shannon entropy, typical sets, and the concentration of measure phenomenon dictate the effective geometry experienced by data distributions. By applying principles from information geometry, we characterize the space of high-dimensional probability distributions as a statistical manifold, where distances and curvatures are defined by information-theoretic metrics like the Fisher information metric. The study reveals that in high dimensions, probability mass concentrates in a thin shell, a counter-intuitive result that has profound implications for statistical inference, machine learning, and data analysis. We demonstrate that the apparent volume of these spaces is governed by entropic quantities, leading to a geometric framework where proximity and structure are naturally expressed in terms of information divergence rather than traditional Euclidean distance. This entropic perspective provides a powerful toolkit for understanding the behavior of algorithms and the intrinsic dimensionality of data in high-dimensional settings. |
| title | Entropic Geometry of High-Dimensional Vector Spaces |
| topic | High-Dimensional Geometry Information Geometry Entropy Concentration of Measure Statistical Manifolds Typical Sets Fisher Information Metric |
| url | https://doi.org/10.5281/zenodo.17455111 |