{"paper":{"title":"Small Ball Probabilities for Simple Random Tensors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Grigoris Paouris, Xuehan Hu","submitted_at":"2024-03-29T14:13:54Z","abstract_excerpt":"We study the small ball probability of an order-$\\ell$ simple random tensor $X=X^{(1)}\\otimes\\cdots\\otimes X^{(\\ell)}$ where $X^{(i)}, 1\\leq i\\leq\\ell$ are independent random vectors in $\\mathbb{R}^n$ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of $X$ onto an $m$-dimensional subspace $F$ falls within an Euclidean ball of length $\\varepsilon$ is upper bounded by $\\frac{\\varepsilon}{(\\ell-1)!}\\left(C\\log\\left(\\frac{e}{\\varepsilon}\\right)\\right)^{\\ell}$ and also this upper bound is sharp when $m$ is small. We also e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.20192","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/2403.20192/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"}