{"paper":{"title":"Minimum Distortion Quantization with Specified Output Distribution","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","math.IT","math.OC","math.ST","stat.TH"],"primary_cat":"cs.IT","authors_text":"Aolin Xu","submitted_at":"2026-06-09T06:06:41Z","abstract_excerpt":"We derive the optimal quantizer of a real-valued random variable $W$ with distribution $P_W$ such that 1) the distribution of the quantization output $X$ that can take $k$ values follows any specified distribution $P_X$ over $\\{1,\\ldots,k\\}$, and 2) the minimum mean squared error (MMSE) of estimating $W$ from $X$ is minimized. It is shown that the optimal quantizer takes the form $X=\\sigma\\big(F_{\\sigma^{-1}(X)}^{-1}(F_W(W))\\big)$, where $\\sigma$ is the optimal permutation of $\\{1,\\ldots,k\\}$ among all permutations to minimize the MMSE, and $F$ is the cumulative distribution function. When $P_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10458","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.10458/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}