{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PCTYEQVHGPQSYULEFCIXHSKBFU","short_pith_number":"pith:PCTYEQVH","canonical_record":{"source":{"id":"1512.04433","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-12-14T17:57:41Z","cross_cats_sorted":["cs.DS","math.FA"],"title_canon_sha256":"242023bf76874051505682e6c6c566471b56765398c2484200688b75e48f3acb","abstract_canon_sha256":"b0ba00e41119279f1e6d907f999909633d2941edb37410aca4d505372a2da6b0"},"schema_version":"1.0"},"canonical_sha256":"78a78242a733e12c5164289173c9412d36e31b4f784dc3597861e3d4ae245d97","source":{"kind":"arxiv","id":"1512.04433","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04433","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04433v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04433","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"PCTYEQVHGPQS","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PCTYEQVHGPQSYULE","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PCTYEQVH","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PCTYEQVHGPQSYULEFCIXHSKBFU","target":"record","payload":{"canonical_record":{"source":{"id":"1512.04433","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-12-14T17:57:41Z","cross_cats_sorted":["cs.DS","math.FA"],"title_canon_sha256":"242023bf76874051505682e6c6c566471b56765398c2484200688b75e48f3acb","abstract_canon_sha256":"b0ba00e41119279f1e6d907f999909633d2941edb37410aca4d505372a2da6b0"},"schema_version":"1.0"},"canonical_sha256":"78a78242a733e12c5164289173c9412d36e31b4f784dc3597861e3d4ae245d97","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:22.262083Z","signature_b64":"BHn/juZ6P8e3Gn/lx7uGkLmM+6xnPw5iNFK3Z4JIcr+EVVUd9UeqS4dpvmf2zUaz1VceuBrdnwoJwyyeH/zcCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78a78242a733e12c5164289173c9412d36e31b4f784dc3597861e3d4ae245d97","last_reissued_at":"2026-05-18T01:24:22.261479Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:22.261479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.04433","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hCV5qMkNnbZbjYjVFT+JFngtP2TjIew0yanJMbzPiLnJCMEXx/kO2u+qPIi6ihIzjysWWXZQGz8KIbCvcfnXBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T00:33:17.157637Z"},"content_sha256":"89e5a423c5352015b46c18d8ddf51548f220b9c65e9fcd0e6978d608bdbee031","schema_version":"1.0","event_id":"sha256:89e5a423c5352015b46c18d8ddf51548f220b9c65e9fcd0e6978d608bdbee031"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PCTYEQVHGPQSYULEFCIXHSKBFU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Near-Optimal Bounds for Binary Embeddings of Arbitrary Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.FA"],"primary_cat":"cs.LG","authors_text":"Ben Recht, Samet Oymak","submitted_at":"2015-12-14T17:57:41Z","abstract_excerpt":"We study embedding a subset $K$ of the unit sphere to the Hamming cube $\\{-1,+1\\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\\omega(K)$ of the set. For subspaces and several structured sets we show that Gaussian maps provide the optimal tradeoff $m\\sim \\delta^{-2}\\omega^2(K)$, in particular for $\\delta$ distortion one needs $m\\approx\\delta^{-2}{d}$ where $d$ is the subspace dimension. For general sets, we provide sharp characterizations which reduces to $m\\approx{\\delta^{-4}}{\\omega^2(K)}$ after simplification. We provide impro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04433","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hWmWgJd3uIyKgPxm7XbTsM4nHB7R4bssZ+mUJ1FZTIFDeRMxu0is/mrMVhHsRGNZG2C1ug1RTVoeVPAAyxqYDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T00:33:17.158096Z"},"content_sha256":"bd4b0c10067050d28db5d7ad6a152d9c5f91498ed18a858ce58557275a11b60c","schema_version":"1.0","event_id":"sha256:bd4b0c10067050d28db5d7ad6a152d9c5f91498ed18a858ce58557275a11b60c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/bundle.json","state_url":"https://pith.science/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-09T00:33:17Z","links":{"resolver":"https://pith.science/pith/PCTYEQVHGPQSYULEFCIXHSKBFU","bundle":"https://pith.science/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/bundle.json","state":"https://pith.science/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PCTYEQVHGPQSYULEFCIXHSKBFU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PCTYEQVHGPQSYULEFCIXHSKBFU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b0ba00e41119279f1e6d907f999909633d2941edb37410aca4d505372a2da6b0","cross_cats_sorted":["cs.DS","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-12-14T17:57:41Z","title_canon_sha256":"242023bf76874051505682e6c6c566471b56765398c2484200688b75e48f3acb"},"schema_version":"1.0","source":{"id":"1512.04433","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04433","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04433v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04433","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"PCTYEQVHGPQS","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PCTYEQVHGPQSYULE","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PCTYEQVH","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:bd4b0c10067050d28db5d7ad6a152d9c5f91498ed18a858ce58557275a11b60c","target":"graph","created_at":"2026-05-18T01:24:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study embedding a subset $K$ of the unit sphere to the Hamming cube $\\{-1,+1\\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\\omega(K)$ of the set. For subspaces and several structured sets we show that Gaussian maps provide the optimal tradeoff $m\\sim \\delta^{-2}\\omega^2(K)$, in particular for $\\delta$ distortion one needs $m\\approx\\delta^{-2}{d}$ where $d$ is the subspace dimension. For general sets, we provide sharp characterizations which reduces to $m\\approx{\\delta^{-4}}{\\omega^2(K)}$ after simplification. We provide impro","authors_text":"Ben Recht, Samet Oymak","cross_cats":["cs.DS","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-12-14T17:57:41Z","title":"Near-Optimal Bounds for Binary Embeddings of Arbitrary Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04433","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:89e5a423c5352015b46c18d8ddf51548f220b9c65e9fcd0e6978d608bdbee031","target":"record","created_at":"2026-05-18T01:24:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b0ba00e41119279f1e6d907f999909633d2941edb37410aca4d505372a2da6b0","cross_cats_sorted":["cs.DS","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-12-14T17:57:41Z","title_canon_sha256":"242023bf76874051505682e6c6c566471b56765398c2484200688b75e48f3acb"},"schema_version":"1.0","source":{"id":"1512.04433","kind":"arxiv","version":1}},"canonical_sha256":"78a78242a733e12c5164289173c9412d36e31b4f784dc3597861e3d4ae245d97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78a78242a733e12c5164289173c9412d36e31b4f784dc3597861e3d4ae245d97","first_computed_at":"2026-05-18T01:24:22.261479Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:22.261479Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BHn/juZ6P8e3Gn/lx7uGkLmM+6xnPw5iNFK3Z4JIcr+EVVUd9UeqS4dpvmf2zUaz1VceuBrdnwoJwyyeH/zcCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:22.262083Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.04433","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89e5a423c5352015b46c18d8ddf51548f220b9c65e9fcd0e6978d608bdbee031","sha256:bd4b0c10067050d28db5d7ad6a152d9c5f91498ed18a858ce58557275a11b60c"],"state_sha256":"3a85fd8562812dff172d9a0612e9c2b51bdec3601b0e64659507328855d62b02"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gdo7vp9P7zohu0tqKftC9ZuQDNf+OZCAv5qFL994F89qNETnJZmKnu8cv5OTL3723H8mvD6pHreG6QOADil0Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T00:33:17.162165Z","bundle_sha256":"18ae4906e2847921fd67e53c0850fa460df227426dfcb44e04d5079be46dc046"}}