{"paper":{"title":"Provable Quantization with Randomized Hadamard Transform","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Dithered quantization after randomized Hadamard transform matches the error of dense random rotations at linearithmic cost.","cross_cats":["cs.DS"],"primary_cat":"cs.LG","authors_text":"Boris Prokhorov, Dmitry Krachun, Michael Kapralov, Piotr Indyk, Ying Feng","submitted_at":"2026-05-13T17:38:18Z","abstract_excerpt":"Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random rotations yield clean theoretical guarantees, they require $\\Theta(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \\log d)$, but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees.\n  In this work, we study a variant of this approach: dithered quantization with a single randomi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A dithered version of TurboQuant achieves mean squared error (π√3/2 + o(1)) · 4^{-b} at b bits per coordinate, where the o(1) term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes the input vectors are unit vectors and relies on the asymptotic regime where the number of quantization levels grows; the uniformity claim over all dimensions and vectors may require additional technical conditions on the dither distribution that are not fully detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Dithered quantization after a single randomized Hadamard transform yields unbiased estimates whose MSE asymptotically equals that of dense random rotations, specifically (π√3/2 + o(1))·4^{-b} for b-bit TurboQuant.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Dithered quantization after randomized Hadamard transform matches the error of dense random rotations at linearithmic cost.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"08c9edaf8d30133c15f7c76c2cbf144db3de9be856b06176d9c757c0b4920b15"},"source":{"id":"2605.13810","kind":"arxiv","version":1},"verdict":{"id":"55cdd0fc-1ed5-43e0-9f5a-eb4363b8a519","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:23:05.736386Z","strongest_claim":"A dithered version of TurboQuant achieves mean squared error (π√3/2 + o(1)) · 4^{-b} at b bits per coordinate, where the o(1) term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.","one_line_summary":"Dithered quantization after a single randomized Hadamard transform yields unbiased estimates whose MSE asymptotically equals that of dense random rotations, specifically (π√3/2 + o(1))·4^{-b} for b-bit TurboQuant.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes the input vectors are unit vectors and relies on the asymptotic regime where the number of quantization levels grows; the uniformity claim over all dimensions and vectors may require additional technical conditions on the dither distribution that are not fully detailed in the abstract.","pith_extraction_headline":"Dithered quantization after randomized Hadamard transform matches the error of dense random rotations at linearithmic cost."},"references":{"count":38,"sample":[{"doi":"","year":null,"title":"Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing , pages=","work_id":"47611ad1-835b-47cc-b583-fd18f46a3525","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"IEEE Transactions on Pattern Analysis and Machine Intelligence , volume=","work_id":"c5b78240-0e0e-4924-9a71-749f762e602a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proceedings of the Twentieth Annual Symposium on Computational Geometry , pages=","work_id":"a37bab56-f417-4ec3-98f3-77e0f250ba56","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proceedings of the 31st annual international ACM SIGIR conference on Research and development in information retrieval , pages=","work_id":"951997c4-4a60-403e-a58f-237476c67aa6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proceedings of the AAAI Conference on Artificial Intelligence , volume=","work_id":"67608266-46ab-4cc0-aaa8-1ebc437a78e5","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"791bbbbbd07ffe8252d6b0c33335e8bdadb69c2f447f3404991b1353239ad2a9","internal_anchors":3},"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"}