{"paper":{"title":"Upper and lower bounds for the Bregman divergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Benjamin Sprung","submitted_at":"2018-08-02T12:03:04Z","abstract_excerpt":"In this paper we study upper and lower bounds on the Bregman divergence $\\Delta_{\\mathcal{F}}^{\\xi}(y,x):=\\mathcal{F}(y)-\\mathcal{F}(x)-\\langle \\xi, y-x\\rangle $ for some convex functional $\\mathcal{F}$ on a normed space $\\mathcal{X}$, with subgradient $\\xi\\in\\partial\\mathcal{F}(x)$. We give a considerably simpler new proof of the inequalities by Xu and Roach for the special case $\\mathcal{F}(x)=\\left\\| x\\right\\|^p, p>1$. The results can be transfered to more general functions as well."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.00772","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":""},"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"}