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Localization costs and information growth for exact Gaussian observations

OpenAI · September 27, 2026

Lean

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Abstract

For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of $\Theta(d)$ exact Gaussian measurements reveal only $O_A(dt)$ information when each message has at most $\exp(Ad^2)$ values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI, Memory–sample lower bounds for noiseless Gaussian regression: Lean scope notes (oai-math-revised mirror of openai/math, 2026-10). Describes the Lean formalization across the six papers, with eight comparator files (e.g. NoiselessRegression.lean, SubsphereCurrent.lean); for this paper it reports repeated-localization information bounds and finite-entropy regularization with almost-sure termination and finite expected depth. [family-level]

Digests and expositions

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Elsewhere

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Cite

@misc{OAI:Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026,
  author = {{OpenAI}},
  title = {{Localization costs and information growth for exact Gaussian observations}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026/paper.pdf}{OAI:Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/140.md. The Lean statements have not been checked by people against the paper's claims.

Family 140: Memory–sample lower bounds for noiseless Gaussian regression

For fixed A > 0, a one-pass learner with $Ad^2$ persistent bits needs $\Omega_A(d\log(1/\epsilon))$ noiseless Gaussian samples to recover a unit vector to angular error $0\lt \epsilon\le1/10$ with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.

Other papers in this family: