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The spherical perceptron with bi-orthogonally invariant disorder
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Abstract
We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
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Possibly relevant literature (found by the agent, not checked)
- Cédric Gerbelot, Alia Abbara, Florent Krzakala, Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or: How to Prove Kabashima's Replica Formula) (arXiv:2006.06581, IEEE Trans. Inf. Theory 69 (2023)). This paper proves rigorously, for convex GLMs with rotationally invariant data matrices of bounded spectrum, a replica formula in the Kabashima framework that this paper cites as a predecessor of its spectral model.
Digests and expositions
- AI Proved What?, How much can one neuron store? Perceptrons and jamming (Episode 222, October 2026). This AI-assisted family overview briefly discusses the structured-disorder extension. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 222
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026,
author = {{OpenAI}},
title = {{The spherical perceptron with bi-orthogonally invariant disorder}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf}{OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/222.md. The Lean statements have not been checked by people against the paper's claims.
Family 222: Perceptron free energies and microscopic jamming exponents
Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.
Other papers in this family:
- The free energy of the Ising random perceptron · September 24, 2026
- Microscopic jamming in the negative spherical perceptron · September 24, 2026
- The free energy of the spherical random perceptron · September 24, 2026