{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:CVZELOXV7OWZB6NXZZIZNMQPES","short_pith_number":"pith:CVZELOXV","canonical_record":{"source":{"id":"1308.2768","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-13T06:41:37Z","cross_cats_sorted":[],"title_canon_sha256":"9ad9df757dab8fbd462daab63d91bb47c4a066ef0ac71777ec48b8e529935db8","abstract_canon_sha256":"f001a89b81b1c9a241383546415eca4628d7e7d02caa7691c53567699bc5373f"},"schema_version":"1.0"},"canonical_sha256":"157245baf5fbad90f9b7ce5196b20f24907874d39d6c8a0ece4af310312904a3","source":{"kind":"arxiv","id":"1308.2768","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.2768","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"arxiv_version","alias_value":"1308.2768v1","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2768","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"pith_short_12","alias_value":"CVZELOXV7OWZ","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CVZELOXV7OWZB6NX","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CVZELOXV","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:CVZELOXV7OWZB6NXZZIZNMQPES","target":"record","payload":{"canonical_record":{"source":{"id":"1308.2768","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-13T06:41:37Z","cross_cats_sorted":[],"title_canon_sha256":"9ad9df757dab8fbd462daab63d91bb47c4a066ef0ac71777ec48b8e529935db8","abstract_canon_sha256":"f001a89b81b1c9a241383546415eca4628d7e7d02caa7691c53567699bc5373f"},"schema_version":"1.0"},"canonical_sha256":"157245baf5fbad90f9b7ce5196b20f24907874d39d6c8a0ece4af310312904a3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:05.298215Z","signature_b64":"HCFSMpxFo5fSaVNHPL/RakWvhggyURT5bIwTlPsf0+dJjCwM+uHrt5EhQ6KmxaA6yy1HKyFbxfvoNqpjDe/kAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"157245baf5fbad90f9b7ce5196b20f24907874d39d6c8a0ece4af310312904a3","last_reissued_at":"2026-05-18T03:16:05.297550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:05.297550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.2768","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":"2026-05-18T03:16:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wnPXlGorbQfW6V1l2QzLSUDNb6aFe5rt//GjWSPXqYR0oR1O9kyS3MVjQBGQ/FXbzyTH78QcYSui1xGdZhlWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:06:46.844704Z"},"content_sha256":"ed4de2b36c4489cdccdd2260f036760a2341510e3716aa2c479192be92748b65","schema_version":"1.0","event_id":"sha256:ed4de2b36c4489cdccdd2260f036760a2341510e3716aa2c479192be92748b65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:CVZELOXV7OWZB6NXZZIZNMQPES","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Large distortion dimension reduction using random variable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alon Dmitriyuk, Yehoram Gordon","submitted_at":"2013-08-13T06:41:37Z","abstract_excerpt":"Consider a random matrix $H:\\mathbb{R}^n\\longrightarrow\\mathbb{R}^m$. Let $D\\geq2$ and let $\\{W_l\\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\\mathbb{R}^n$. We ask what is the probability that for all $1\\leq l\\leq p$ and $x,y\\in W_l$, \\[\n  \\|x-y\\|_2\\leq\\|Hx-Hy\\|_2\\leq D\\|x-y\\|_2. \\] We show that for $m=O\\big(k+\\frac{\\ln{p}}{\\ln{D}}\\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2768","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":"2026-05-18T03:16:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/+r4c5FWbcObt3oE79wiAawtIxujoHo7BdJKFgUMVkxHUqXehywhQn1lXO9RK40hoffpRUOxCXTl/RAnmmUnBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:06:46.845413Z"},"content_sha256":"3c9b91db7072fc40c9213e1b74e7952e411b397b445b12ab316fc63c30ff28b1","schema_version":"1.0","event_id":"sha256:3c9b91db7072fc40c9213e1b74e7952e411b397b445b12ab316fc63c30ff28b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CVZELOXV7OWZB6NXZZIZNMQPES/bundle.json","state_url":"https://pith.science/pith/CVZELOXV7OWZB6NXZZIZNMQPES/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CVZELOXV7OWZB6NXZZIZNMQPES/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":"2026-06-08T18:06:46Z","links":{"resolver":"https://pith.science/pith/CVZELOXV7OWZB6NXZZIZNMQPES","bundle":"https://pith.science/pith/CVZELOXV7OWZB6NXZZIZNMQPES/bundle.json","state":"https://pith.science/pith/CVZELOXV7OWZB6NXZZIZNMQPES/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CVZELOXV7OWZB6NXZZIZNMQPES/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:CVZELOXV7OWZB6NXZZIZNMQPES","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":"f001a89b81b1c9a241383546415eca4628d7e7d02caa7691c53567699bc5373f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-13T06:41:37Z","title_canon_sha256":"9ad9df757dab8fbd462daab63d91bb47c4a066ef0ac71777ec48b8e529935db8"},"schema_version":"1.0","source":{"id":"1308.2768","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.2768","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"arxiv_version","alias_value":"1308.2768v1","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2768","created_at":"2026-05-18T03:16:05Z"},{"alias_kind":"pith_short_12","alias_value":"CVZELOXV7OWZ","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CVZELOXV7OWZB6NX","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CVZELOXV","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:3c9b91db7072fc40c9213e1b74e7952e411b397b445b12ab316fc63c30ff28b1","target":"graph","created_at":"2026-05-18T03:16:05Z","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":"Consider a random matrix $H:\\mathbb{R}^n\\longrightarrow\\mathbb{R}^m$. Let $D\\geq2$ and let $\\{W_l\\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\\mathbb{R}^n$. We ask what is the probability that for all $1\\leq l\\leq p$ and $x,y\\in W_l$, \\[\n  \\|x-y\\|_2\\leq\\|Hx-Hy\\|_2\\leq D\\|x-y\\|_2. \\] We show that for $m=O\\big(k+\\frac{\\ln{p}}{\\ln{D}}\\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$.","authors_text":"Alon Dmitriyuk, Yehoram Gordon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-13T06:41:37Z","title":"Large distortion dimension reduction using random variable"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2768","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:ed4de2b36c4489cdccdd2260f036760a2341510e3716aa2c479192be92748b65","target":"record","created_at":"2026-05-18T03:16:05Z","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":"f001a89b81b1c9a241383546415eca4628d7e7d02caa7691c53567699bc5373f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-08-13T06:41:37Z","title_canon_sha256":"9ad9df757dab8fbd462daab63d91bb47c4a066ef0ac71777ec48b8e529935db8"},"schema_version":"1.0","source":{"id":"1308.2768","kind":"arxiv","version":1}},"canonical_sha256":"157245baf5fbad90f9b7ce5196b20f24907874d39d6c8a0ece4af310312904a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"157245baf5fbad90f9b7ce5196b20f24907874d39d6c8a0ece4af310312904a3","first_computed_at":"2026-05-18T03:16:05.297550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:16:05.297550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HCFSMpxFo5fSaVNHPL/RakWvhggyURT5bIwTlPsf0+dJjCwM+uHrt5EhQ6KmxaA6yy1HKyFbxfvoNqpjDe/kAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:16:05.298215Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.2768","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed4de2b36c4489cdccdd2260f036760a2341510e3716aa2c479192be92748b65","sha256:3c9b91db7072fc40c9213e1b74e7952e411b397b445b12ab316fc63c30ff28b1"],"state_sha256":"475145971d373813a07d3592276589390ce6ced2e893fb6c0567bcf600474df4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UERgVmyfquPY8ToZA57cqL44fSu7KVZ76aTKklxMuZTCZSeJ+H7tBIJDDTDgeY+/Bu4DA9k5z3o0z8f+iZ2uCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:06:46.849399Z","bundle_sha256":"897ab062a71cc41a7e1d91d23399dcfed219364e4427dee712bb257561817ff7"}}