{"paper":{"title":"Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Nutan Limaye, Srikanth Srinivasan, Suryajith Chillara","submitted_at":"2017-10-16T03:01:52Z","abstract_excerpt":"In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\\times 2$ matrices, denoted $\\mathrm{IMM}_{d}$, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear.\n  Formally, for each depth $\\Delta \\leq \\log d$, we show that any product-depth $\\Delta$ multilinear formula for $\\mathrm{IMM}_d$ must have size $\\exp(\\Omega(\\Delta d^{1/\\Delta})).$ It also follows from this that any multilinear circuit of product-depth $\\Delta$ for the same polynomial of the above form must have a size o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05481","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"}