{"id":"openai-math/Posterior-replicas-and-conditional-information-in-Gaussian-regression","pdf":"https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026","tags":["information-theory","statistics","learning-theory"],"score":0,"title":"Posterior replicas and conditional information in Gaussian regression","domain":"Theoretical computer science","family":"140","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"cs.CC","summary":"For a signal with density bounded by L relative to uniform probability on $`S^{d-1}`$, we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d , the bound is $`O(H(W)/d+d+\\log(2+\\log L))`$. Consequently, a finite-state learner with $`o(d^2)`$ persistent bits and a deterministic sample horizon needs $`\\Omega(d\\log(1/\\epsilon))`$ fresh noiseless Gaussian measurements for constant-probability angular accuracy $`0\\lt \\epsilon\\le1/10`$ under the uniform…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":true,"family_title":"Memory–sample lower bounds for noiseless Gaussian regression"}