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Since such measurements give no information on the norm of ${x}$, recovery methods from such measurements typically assume that $\\| {x} \\|_2=1$. We show that if one allows more generally for quantized affine measurements of the form $\\operatorname{sign}(\\langle {a}_i, {x} \\rangle + b_i)$, and if the vectors ${a}_i$ are "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1404.6853","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2014-04-28T02:01:33Z","cross_cats_sorted":["math.NA","math.OC","math.PR"],"title_canon_sha256":"ba83050b28fc63f7f079a02418e819ac74577f3d8e29790c27bb396849bf05eb","abstract_canon_sha256":"e221542c8bf55359ac4994062d3dbcddbfde58603a724e70281ba8921ff710c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:43.659283Z","signature_b64":"1b6HocNEHihmsp0+V3S0VEUZSMgI9so+nQdIKO7RwA5sJt/PqDF3JlISNny7j+2x1IJf2utcO3yuOhAqZ2fwDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"709d072b5bdb50101d3c1ab49f2511cd4a7b433fb5a90b3bdd29f9ee87c76af9","last_reissued_at":"2026-05-18T01:22:43.658647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:43.658647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"One-bit compressive sensing with norm estimation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA","math.OC","math.PR"],"primary_cat":"stat.ML","authors_text":"Karin Knudson, Rachel Ward, Rayan Saab","submitted_at":"2014-04-28T02:01:33Z","abstract_excerpt":"Consider the recovery of an unknown signal ${x}$ from quantized linear measurements. 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