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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 $P_W$ is uniform over an interval or $P_X$ is uniform over $\\{1,\\ldots,k\\}$, the quantizer takes a simple form $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information betwe","title":"Minimum Distortion Quantization with Specified Output Distribution","url":"https://arxiv.org/abs/2606.10458","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.10458v1 Announce Type: cross \nAbstract: 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 $P_W$ is uniform over an interval or $P_X$ is uniform over $\\{1,\\ldots,k\\}$, the quantizer takes a simple form $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information betwe","title":"Minimum Distortion Quantization with Specified Output Distribution","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-10T04:43:37Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.10458"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:0ad5d45d377ebf30f86cf26ba889b6cd7842e6f7409b631b98b0047d997b5531f6ae35a2cc71050b049537b62d68b46e5e198e26b80be6524c2b224c396da907","signer":"crovia.substrate","subject":{"observed_at":"2026-06-10T04:43:37Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.10458"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"23cc2fab8b4b187aa6b6d2ad6ca2da8ba9ed0eab1740bade8969d056824636e0","leaf_index":226032,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"3ad440984cacf7627d0be616ae85459b88c2d49e538af33fdb03dc7003883472","side":"right"},{"sibling":"9b32d1a15bcd4c89a740f720c3dbdbaf1191a3160e131dd4571ccdcf7ab2d6df","side":"right"},{"sibling":"f3db8a57151187396c9ea86d53296b625736f05dc1226a1416c7a45e4bbb0176","side":"right"},{"sibling":"c4b7acb535f2691f55096391c309cc0ede8bca67c5aa552993ca136c0148f023","side":"right"},{"sibling":"878f7509e9790121ef4e4fda20a3573a3c86fe2c314a0fe2d24e44153fac2409","side":"left"},{"sibling":"1827bc96175581d6a3b558cd294f2fa681de8e748ad51bcefabf97b967578932","side":"left"},{"sibling":"842f29cae12eabb338c138a303020275424d27ef5b0463ac7e8af799b137bc4a","side":"left"},{"sibling":"3d9f311f467db31ee1be2d9664daf3ef6547a3814a86217d383241d0655ab769","side":"left"},{"sibling":"ac698c3a6027f35b513fe892f166e70344e855625238cbb581d0bf06c131db80","side":"right"},{"sibling":"a4d17aefe58175050dc159af6246658fcf1c9f3ed57aacf1b350fc3261de4e69","side":"left"},{"sibling":"280b980aa0c7756b0b0cb22658f26466d36f0e70fbc3312cd2311d9898e30b8f","side":"right"},{"sibling":"c98954d4b658b1dda60fe52576fcf9bf21a2d49c67fb63f8c30f16ab5f721938","side":"right"},{"sibling":"cdb58f86163046d3b15f857b03372ec75e1ad9ea4548e086793d528b9eed364d","side":"left"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"6cea4964f32722eb370847c2f7c9d6a9f0622c239538b07e6815a59d6fd8d49c","side":"right"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":228173,"merkle_root":"7e416202c0bfd759bd2eea4236713b403993d99793fe8badb5065040080bece3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260611T143708Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-11T21:59:35Z","sig_algorithm":"ed25519","signature":"231c80024bc3982dd493c45b31af95097e97aabc6d712a4e5bad7d0cbdd3c08e01ff395b0f8e72754bac97016e0cd0eed88b8a13cb71edbbcb9b6d72c10a7b03","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_500312889532d7e3b1ea85e637e674c59cd9cfa30de2186b96b2d42a37049450"}}