{"paper":{"title":"The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.GT","cs.LG","math.PR","math.ST","stat.TH"],"primary_cat":"cs.DS","authors_text":"Alistair Stewart, Daniel M. Kane, Ilias Diakonikolas","submitted_at":"2015-11-11T18:00:37Z","abstract_excerpt":"An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \\sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is {\\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications:\n  {\\bf Learning Theory.} We de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03592","kind":"arxiv","version":2},"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"}