{"paper":{"title":"Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.NE"],"primary_cat":"cs.CC","authors_text":"R. Ryan Williams","submitted_at":"2018-02-26T01:30:21Z","abstract_excerpt":"We consider the problem of representing Boolean functions exactly by \"sparse\" linear combinations (over $\\mathbb{R}$) of functions from some \"simple\" class ${\\cal C}$. In particular, given ${\\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\\cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${\\cal C}$ is \"overcomplete\" and the set of functions is not linearly independent. We focus on the cases where ${\\cal C}$ is the set of linear threshol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09121","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"}