{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:3XT3LVCMUAILCE57SUCVTQNDDM","short_pith_number":"pith:3XT3LVCM","canonical_record":{"source":{"id":"1107.0992","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-05T20:49:59Z","cross_cats_sorted":[],"title_canon_sha256":"a15369f8eefde2074d86b4214f9acf508465eeef05754ee17dffbeb4d23d6fc0","abstract_canon_sha256":"ca3dc58d7d7f693564ed93ec37de84ce2b4b207dd7ce79a8293a272a1067890e"},"schema_version":"1.0"},"canonical_sha256":"dde7b5d44ca010b113bf950559c1a31b0620af6cc0624bd5fba174dc02fbe582","source":{"kind":"arxiv","id":"1107.0992","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.0992","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"arxiv_version","alias_value":"1107.0992v1","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.0992","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"pith_short_12","alias_value":"3XT3LVCMUAIL","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"3XT3LVCMUAILCE57","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"3XT3LVCM","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:3XT3LVCMUAILCE57SUCVTQNDDM","target":"record","payload":{"canonical_record":{"source":{"id":"1107.0992","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-05T20:49:59Z","cross_cats_sorted":[],"title_canon_sha256":"a15369f8eefde2074d86b4214f9acf508465eeef05754ee17dffbeb4d23d6fc0","abstract_canon_sha256":"ca3dc58d7d7f693564ed93ec37de84ce2b4b207dd7ce79a8293a272a1067890e"},"schema_version":"1.0"},"canonical_sha256":"dde7b5d44ca010b113bf950559c1a31b0620af6cc0624bd5fba174dc02fbe582","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:45.874691Z","signature_b64":"PczHRu5SjJ60ZgsmEicY6v1GEzAEhBhnmVfTC3/WiFFdsrExJqkNM7NOLPrJMhxGW8Mx32jzXJxBpBQxEo//Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dde7b5d44ca010b113bf950559c1a31b0620af6cc0624bd5fba174dc02fbe582","last_reissued_at":"2026-05-18T04:18:45.874028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:45.874028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.0992","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":"2026-05-18T04:18:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+8gn7+PkClZKigyBi4Iz1kP6M8qLiVZz6N8nwg7pGQHEe6IueBvYUV6By7SyfbZ1nMwdCbAjt3jmpzdAZ/TBAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T05:29:02.334804Z"},"content_sha256":"6927c31bfc386e9b1d0d0cb3d2eeaa3d5323d41a295c423d653c2317bd349880","schema_version":"1.0","event_id":"sha256:6927c31bfc386e9b1d0d0cb3d2eeaa3d5323d41a295c423d653c2317bd349880"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:3XT3LVCMUAILCE57SUCVTQNDDM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sparsity and non-Euclidean embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Olivier Gu\\'edon, Omer Friedland","submitted_at":"2011-07-05T20:49:59Z","abstract_excerpt":"We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \\le 1$, we construct a family of operators that embed $\\ell_p^n$ into $\\ell_r^{(1+\\eta)n}$, with optimal polynomial bounds in $\\eta >0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0992","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":"2026-05-18T04:18:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K2to6B+It2GVJqVYBAEtEKxjwjksPgBtjjm2KWQQqU8qiZDqeXi2q03hp6Q7/gXY6N/j/HtQbTPCzTpjuKNlAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T05:29:02.335147Z"},"content_sha256":"c99eabc5c2e772fab73b097c405ae9fdc45f1316edc96201030dc673c4895c23","schema_version":"1.0","event_id":"sha256:c99eabc5c2e772fab73b097c405ae9fdc45f1316edc96201030dc673c4895c23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3XT3LVCMUAILCE57SUCVTQNDDM/bundle.json","state_url":"https://pith.science/pith/3XT3LVCMUAILCE57SUCVTQNDDM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3XT3LVCMUAILCE57SUCVTQNDDM/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":"2026-06-25T05:29:02Z","links":{"resolver":"https://pith.science/pith/3XT3LVCMUAILCE57SUCVTQNDDM","bundle":"https://pith.science/pith/3XT3LVCMUAILCE57SUCVTQNDDM/bundle.json","state":"https://pith.science/pith/3XT3LVCMUAILCE57SUCVTQNDDM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3XT3LVCMUAILCE57SUCVTQNDDM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3XT3LVCMUAILCE57SUCVTQNDDM","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":"ca3dc58d7d7f693564ed93ec37de84ce2b4b207dd7ce79a8293a272a1067890e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-05T20:49:59Z","title_canon_sha256":"a15369f8eefde2074d86b4214f9acf508465eeef05754ee17dffbeb4d23d6fc0"},"schema_version":"1.0","source":{"id":"1107.0992","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.0992","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"arxiv_version","alias_value":"1107.0992v1","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.0992","created_at":"2026-05-18T04:18:45Z"},{"alias_kind":"pith_short_12","alias_value":"3XT3LVCMUAIL","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"3XT3LVCMUAILCE57","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"3XT3LVCM","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:c99eabc5c2e772fab73b097c405ae9fdc45f1316edc96201030dc673c4895c23","target":"graph","created_at":"2026-05-18T04:18:45Z","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 present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \\le 1$, we construct a family of operators that embed $\\ell_p^n$ into $\\ell_r^{(1+\\eta)n}$, with optimal polynomial bounds in $\\eta >0$.","authors_text":"Olivier Gu\\'edon, Omer Friedland","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-05T20:49:59Z","title":"Sparsity and non-Euclidean embeddings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.0992","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":"sha256:6927c31bfc386e9b1d0d0cb3d2eeaa3d5323d41a295c423d653c2317bd349880","target":"record","created_at":"2026-05-18T04:18:45Z","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":"ca3dc58d7d7f693564ed93ec37de84ce2b4b207dd7ce79a8293a272a1067890e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-05T20:49:59Z","title_canon_sha256":"a15369f8eefde2074d86b4214f9acf508465eeef05754ee17dffbeb4d23d6fc0"},"schema_version":"1.0","source":{"id":"1107.0992","kind":"arxiv","version":1}},"canonical_sha256":"dde7b5d44ca010b113bf950559c1a31b0620af6cc0624bd5fba174dc02fbe582","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dde7b5d44ca010b113bf950559c1a31b0620af6cc0624bd5fba174dc02fbe582","first_computed_at":"2026-05-18T04:18:45.874028Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:45.874028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PczHRu5SjJ60ZgsmEicY6v1GEzAEhBhnmVfTC3/WiFFdsrExJqkNM7NOLPrJMhxGW8Mx32jzXJxBpBQxEo//Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:45.874691Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.0992","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6927c31bfc386e9b1d0d0cb3d2eeaa3d5323d41a295c423d653c2317bd349880","sha256:c99eabc5c2e772fab73b097c405ae9fdc45f1316edc96201030dc673c4895c23"],"state_sha256":"ba8c74e6764b4d366f78324bfad2e1b8c786d813068e92c255d54ebd9572b57d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W49AR5MaRf0c4kopClS43vRnPsMFNj0Fd3qy4yXNc7hcoPgTKhOhFHXtETCyGUf8BtD7T0PXFg4dWVsesXJTCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T05:29:02.337099Z","bundle_sha256":"d278d5e58784c987f01057c08696acd225e6083ac44a38637a9794180854bc0f"}}