Learning reshapes power-law anisotropy in internal representations

[2608.15239] Learning reshapes power-law anisotropy in internal representations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2608.15239 (cs) [Submitted on 15 Aug 2026] Title: Learning reshapes power-law anisotropy in internal representations Authors: Asahi Nakamuta , Jun-nosuke Teramae View a PDF of the paper titled Learning reshapes power-law anisotropy in internal representations, by Asahi Nakamuta and Jun-nosuke Teramae View PDF HTML (experimental) Abstract: Power-law anisotropy in internal representations has been observed across a wide range of biological and artificial neural systems, from state-of-the-art language models to the mouse cerebral cortex. This anisotropy is a key geometric property of high-dimensional information processing and underlies a variety of theoretical analyses. However, the mechanism by which it emerges from input structure and task-driven learning has remained unclear. Here, we characterize this formation process by exactly solving the learning dynamics of a wide two-layer linear neural network in a teacher--student setting with power-law input and teacher structures. We show that, in the feature-learning regime, the local power-law exponent of the internal-representation spectrum evolves nonmonotonically over the course of training and exhibits up to four distinct asymptotic regimes across modes and training times. By contrast, in the lazy regime, the exponent remains essentially unchanged. We further demonstrate numerically that similar exponent dynamics arise in more realistic nonlinear networks. Together, these results suggest a general mechanism by which the dynamic interaction between input statistics and task structure gives rise to power-law internal representations. Comments: 26 pages, 6 figures Subjects: Machine Learning (cs.LG) ; Machine Learning (stat.ML) Cite as: arXiv:2608.15239 [cs.LG] (or arXiv:2608.15239v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2608.15239 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Asahi Nakamuta [ view email ] [v1] Sat, 15 Aug 2026 13:56:28 UTC (2,079 KB) Full-text links: Access Paper: View a PDF of the paper titled Learning reshapes power-law anisotropy in internal representations, by Asahi Nakamuta and Jun-nosuke Teramae View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG < prev | next > new | recent | 2026-08 Change to browse by: cs stat stat.ML References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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