{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:2S3W7OKJZVIY2QQJQ2JGUSGYUU","short_pith_number":"pith:2S3W7OKJ","canonical_record":{"source":{"id":"1001.0041","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-12-30T22:48:35Z","cross_cats_sorted":["cs.CC","math.FA"],"title_canon_sha256":"b022ef4c8cba08ae81851156fc5a99920d04c45784d12de6c0126b8e0fa13203","abstract_canon_sha256":"2f1c7f3bd10b9c4dc4895f350d33d513eeed7be7f3afb2d78e3198c3f898ffc2"},"schema_version":"1.0"},"canonical_sha256":"d4b76fb949cd518d420986926a48d8a512d321fca3f9c2d76933d44f13475739","source":{"kind":"arxiv","id":"1001.0041","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0041","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0041v2","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0041","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"pith_short_12","alias_value":"2S3W7OKJZVIY","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"2S3W7OKJZVIY2QQJ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"2S3W7OKJ","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:2S3W7OKJZVIY2QQJQ2JGUSGYUU","target":"record","payload":{"canonical_record":{"source":{"id":"1001.0041","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-12-30T22:48:35Z","cross_cats_sorted":["cs.CC","math.FA"],"title_canon_sha256":"b022ef4c8cba08ae81851156fc5a99920d04c45784d12de6c0126b8e0fa13203","abstract_canon_sha256":"2f1c7f3bd10b9c4dc4895f350d33d513eeed7be7f3afb2d78e3198c3f898ffc2"},"schema_version":"1.0"},"canonical_sha256":"d4b76fb949cd518d420986926a48d8a512d321fca3f9c2d76933d44f13475739","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:05.148880Z","signature_b64":"0l2O5eAeHQm04MMPYdxSrfjNVftrqBaQxDTi9TedpClxl4pkBVWz1ZlVmgwj3DyyhHRIuwIFJupc0UcHGtooAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4b76fb949cd518d420986926a48d8a512d321fca3f9c2d76933d44f13475739","last_reissued_at":"2026-05-18T04:41:05.148451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:05.148451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.0041","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LdOI3QQ2TcitwOd5YR5Pe162z4UYNAk5M5dM5oduzjsaxCtykEiXp0ukArXp4rrIUe49oawrnjxkcQFBOvv1Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T05:29:04.799397Z"},"content_sha256":"556ace03375ac75567e8de2205d18e9337f73b466211994b778d9181cb2b8848","schema_version":"1.0","event_id":"sha256:556ace03375ac75567e8de2205d18e9337f73b466211994b778d9181cb2b8848"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:2S3W7OKJZVIY2QQJQ2JGUSGYUU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost-Euclidean subspaces of $\\ell_1^N$ via tensor products: a simple approach to randomness reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.FA"],"primary_cat":"math.MG","authors_text":"Piotr Indyk, Stanislaw Szarek","submitted_at":"2009-12-30T22:48:35Z","abstract_excerpt":"It has been known since 1970's that the N-dimensional $\\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a \"low-tech\" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\\Omega(N)$-dimensional subspaces of $\\ell_1^N$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0041","kind":"arxiv","version":2},"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":"2026-05-18T04:41:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fwRRcFTvNfJ3O/FR0z8fHesjdHC86G4raSqRMkDjO4Gjc4RYX/pmLdujLFuNVjzWQlnqMIhQv3d686CjGQoSDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T05:29:04.799742Z"},"content_sha256":"a42cb4b385935e714e33178af0e132a3bb248e9364d5a9bbcb3ddc75b8747715","schema_version":"1.0","event_id":"sha256:a42cb4b385935e714e33178af0e132a3bb248e9364d5a9bbcb3ddc75b8747715"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/bundle.json","state_url":"https://pith.science/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/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":"2026-06-29T05:29:04Z","links":{"resolver":"https://pith.science/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU","bundle":"https://pith.science/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/bundle.json","state":"https://pith.science/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2S3W7OKJZVIY2QQJQ2JGUSGYUU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:2S3W7OKJZVIY2QQJQ2JGUSGYUU","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":"2f1c7f3bd10b9c4dc4895f350d33d513eeed7be7f3afb2d78e3198c3f898ffc2","cross_cats_sorted":["cs.CC","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-12-30T22:48:35Z","title_canon_sha256":"b022ef4c8cba08ae81851156fc5a99920d04c45784d12de6c0126b8e0fa13203"},"schema_version":"1.0","source":{"id":"1001.0041","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.0041","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"arxiv_version","alias_value":"1001.0041v2","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.0041","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"pith_short_12","alias_value":"2S3W7OKJZVIY","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"2S3W7OKJZVIY2QQJ","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"2S3W7OKJ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a42cb4b385935e714e33178af0e132a3bb248e9364d5a9bbcb3ddc75b8747715","target":"graph","created_at":"2026-05-18T04:41:05Z","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":"It has been known since 1970's that the N-dimensional $\\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a \"low-tech\" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\\Omega(N)$-dimensional subspaces of $\\ell_1^N$","authors_text":"Piotr Indyk, Stanislaw Szarek","cross_cats":["cs.CC","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-12-30T22:48:35Z","title":"Almost-Euclidean subspaces of $\\ell_1^N$ via tensor products: a simple approach to randomness reduction"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0041","kind":"arxiv","version":2},"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":"sha256:556ace03375ac75567e8de2205d18e9337f73b466211994b778d9181cb2b8848","target":"record","created_at":"2026-05-18T04:41:05Z","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":"2f1c7f3bd10b9c4dc4895f350d33d513eeed7be7f3afb2d78e3198c3f898ffc2","cross_cats_sorted":["cs.CC","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-12-30T22:48:35Z","title_canon_sha256":"b022ef4c8cba08ae81851156fc5a99920d04c45784d12de6c0126b8e0fa13203"},"schema_version":"1.0","source":{"id":"1001.0041","kind":"arxiv","version":2}},"canonical_sha256":"d4b76fb949cd518d420986926a48d8a512d321fca3f9c2d76933d44f13475739","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4b76fb949cd518d420986926a48d8a512d321fca3f9c2d76933d44f13475739","first_computed_at":"2026-05-18T04:41:05.148451Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:05.148451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0l2O5eAeHQm04MMPYdxSrfjNVftrqBaQxDTi9TedpClxl4pkBVWz1ZlVmgwj3DyyhHRIuwIFJupc0UcHGtooAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:05.148880Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.0041","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:556ace03375ac75567e8de2205d18e9337f73b466211994b778d9181cb2b8848","sha256:a42cb4b385935e714e33178af0e132a3bb248e9364d5a9bbcb3ddc75b8747715"],"state_sha256":"74cf24c00cf338ee3e3292c259043babafe1623d0e7ed44f376b1a3c7ae68d27"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iwjr3NG0nlGgl2wklqptzypeXMbmlA4OiIbouYQylSkNpn6OSrz4cs8xGK1ZoXYewGmv9PhV57pQ2p67jLf+Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T05:29:04.801577Z","bundle_sha256":"048344a0fcde9190dfd38458562cf690d1c568d3217c731e95b919271ffe785e"}}