{"paper":{"title":"Lower Bounds for Approximate Sign Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Every sign matrix of approximate sign-rank d contains a monochromatic rectangle of size d to the minus O(d) by d to the minus O(d squared).","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Hamed Hatami, Hasti Karimi, Riju Bindua, Robert Robere","submitted_at":"2026-05-01T19:05:06Z","abstract_excerpt":"We prove new upper and lower bounds on $\\epsilon$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \\times n$ sign matrix with approximate sign-rank $d$ contains a monochromatic rectangle of size $d^{-O(d)}m \\times d^{-O(d^2)}n$, paralleling classical results for exact sign-rank. As an application, we establish a lower bound of $\\Omega(\\sqrt{d/\\log d})$ on the $\\epsilon$-approximate sign-rank of large-margin $d$-dimensional half-spaces. Prior to our work, the only general lower bound technique known for approximate s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Every m × n sign matrix with ε-approximate sign-rank d contains a monochromatic rectangle of size d^{-O(d)} m × d^{-O(d²)} n, and as a consequence the ε-approximate sign-rank of large-margin d-dimensional half-spaces is Ω(√(d / log d)).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The points are in general position in R^d, and the Forster-Barthe isotropic position theorem together with the Bourgain-Tzafriri restricted invertibility principle can be applied without additional loss factors that would collapse the d^{-O(d)} subset sizes.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"New lower bounds of Ω(√(d/log d)) for ε-approximate sign-rank of large-margin d-dimensional half-spaces, supported by a geometric theorem guaranteeing large subsets with no common splitting hyperplane.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every sign matrix of approximate sign-rank d contains a monochromatic rectangle of size d to the minus O(d) by d to the minus O(d squared).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fd5fe1fe7b1f1948840354fdf05d45298439277139567d73bfca947f713914bd"},"source":{"id":"2605.01038","kind":"arxiv","version":2},"verdict":{"id":"8185dfa8-4bd4-4d83-8d5e-d1598e114c6e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T14:25:28.476968Z","strongest_claim":"Every m × n sign matrix with ε-approximate sign-rank d contains a monochromatic rectangle of size d^{-O(d)} m × d^{-O(d²)} n, and as a consequence the ε-approximate sign-rank of large-margin d-dimensional half-spaces is Ω(√(d / log d)).","one_line_summary":"New lower bounds of Ω(√(d/log d)) for ε-approximate sign-rank of large-margin d-dimensional half-spaces, supported by a geometric theorem guaranteeing large subsets with no common splitting hyperplane.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The points are in general position in R^d, and the Forster-Barthe isotropic position theorem together with the Bourgain-Tzafriri restricted invertibility principle can be applied without additional loss factors that would collapse the d^{-O(d)} subset sizes.","pith_extraction_headline":"Every sign matrix of approximate sign-rank d contains a monochromatic rectangle of size d to the minus O(d) by d to the minus O(d squared)."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.01038/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T18:39:30.294171Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T17:41:00.971639Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e0cc4140ad9d48f91d776a02ce164d9dd41bb2f5719001cad8f193732fc6396c"},"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"}