{"paper":{"title":"On Multilinear Forms: Bias, Correlation, and Tensor Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Abhishek Bhrushundi, Mrinal Kumar, Pooya Hatami, Prahladh Harsha, Swastik Kopparty","submitted_at":"2018-04-24T16:35:53Z","abstract_excerpt":"In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \\{0,1\\}$. We show the following results for multilinear forms and tensors.\n  1. Correlation bounds : We show that a random $d$-linear form has exponentially low correlation with low-degree polynomials. More precisely, for $d \\ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \\dots, X_d) : \\left(GF(2)^{k}\\right)^d \\rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$ with any polynomial of degree at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09124","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"}