{"paper":{"title":"A Tutorial on Bregman Projection in Statistics","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Gunhee Cho, Jae Kwang Kim, Yumou Qiu","submitted_at":"2026-06-19T19:59:44Z","abstract_excerpt":"A single geometric operation -- projecting a reference onto a constrained family under a Bregman divergence -- underlies a striking range of statistical methods. This tutorial develops the operation first as pure convex geometry, with no statistics attached. A strictly convex generator $G$ and its conjugate $F$ furnish two coordinate systems, a projection theorem with existence and uniqueness, and a Pythagorean {theorem}; the Pythagorean theorem itself produces {two} dual projections -- the information (e-) projection onto moment-constrained families and the moment (m-) projection onto exponen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21714","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/2606.21714/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"}