{"paper":{"title":"Non-asymptotic quantisation of spherically symmetric distributions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For spherically symmetric distributions, random quantizers placed uniformly on a sphere of optimal radius achieve low expected distortion even with moderate numbers of points in high dimensions.","cross_cats":["math.PR","stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Anatoly Zhigljavsky, Luc Pronzato","submitted_at":"2026-05-12T10:01:41Z","abstract_excerpt":"Zador's celebrated theorem is a cornerstone of optimal quantisation, establishing both the weak limit of the empirical distribution of an $n$-point optimal quantiser in $R^d$ and the decay rate of the associated $L_s$-mean quantisation error. However, for large dimensions $d$, observing this asymptotic behaviour demands an astronomically large sample size $n$, which grows super-exponentially with $d$. Through a detailed analysis of the quantisation problem for spherically symmetric distributions, we demonstrate that for moderate $n$ random quantisers uniformly distributed on a sphere of suitab"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For spherically symmetric distributions, random quantisers uniformly distributed on a sphere of suitable radius r achieve exceptional performance; the expected distortion is a triple integral computable with arbitrary precision, and the optimal r can be determined numerically, with approximations from extreme-value theory when n scales with d.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The distributions under study are spherically symmetric, and the claimed performance advantage holds specifically for moderate n where the asymptotic regime of Zador's theorem has not yet been reached.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For spherically symmetric distributions, random points on an optimally chosen sphere achieve low expected distortion for moderate n, with radius approximations derived from extreme value theory.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For spherically symmetric distributions, random quantizers placed uniformly on a sphere of optimal radius achieve low expected distortion even with moderate numbers of points in high dimensions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"83b7faa2bb28b94b4a3e0fe3b90c286004f077ca083cbeb07f1444b4b815d3d3"},"source":{"id":"2605.12568","kind":"arxiv","version":1},"verdict":{"id":"36a67f2e-cb5b-4c7d-ac4b-c6cf70fe6482","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:47:50.672722Z","strongest_claim":"For spherically symmetric distributions, random quantisers uniformly distributed on a sphere of suitable radius r achieve exceptional performance; the expected distortion is a triple integral computable with arbitrary precision, and the optimal r can be determined numerically, with approximations from extreme-value theory when n scales with d.","one_line_summary":"For spherically symmetric distributions, random points on an optimally chosen sphere achieve low expected distortion for moderate n, with radius approximations derived from extreme value theory.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The distributions under study are spherically symmetric, and the claimed performance advantage holds specifically for moderate n where the asymptotic regime of Zador's theorem has not yet been reached.","pith_extraction_headline":"For spherically symmetric distributions, random quantizers placed uniformly on a sphere of optimal radius achieve low expected distortion even with moderate numbers of points in high dimensions."},"references":{"count":13,"sample":[{"doi":"","year":2000,"title":"N. Alon and J.H. Spencer.The Probabilistic Method. Wiley, 2000. Second edition","work_id":"3f9c83e2-7502-4b44-b688-4961b43fbb1c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"K.-T. Fang, S. Kotz, and K.W. Ng.Symmetric Multivariate and Related Dis- tributions. Chapman and Hall/CRC, 1990","work_id":"c2f8edad-a2de-465b-b47e-55fd746aee5d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1971,"title":"Feller.An Introduction to Probability Theory and Its Applications, vol","work_id":"78cc0135-8c0a-4928-82d3-943efa17dc5a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"S. Graf and H. Luschgy.Foundations of Quantization for Probability Distribu- tions. Springer, Berlin, 2000","work_id":"b36e39d3-0725-46b1-a335-8959d70ee81c","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"S. Kotz and S. Nadarajah.Extreme Value Distributions: Theory and Applica- tions. Imperial College Press, 2000","work_id":"b7e03f59-feb1-47e0-aff0-e9ac12780d34","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"4e0c5fb9f69d369c300c1841e0c0d0a3c38ae7d1606a6d255d879e36df1ff886","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"86b58b1c29bf2b9da9ac0c1d159bc0d0d5b8db54e8aa2cc0f36d8d568f4e4e55"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}