{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:NE365RFB4TYAIYH72BZA6ATUVR","short_pith_number":"pith:NE365RFB","canonical_record":{"source":{"id":"1904.07763","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2019-04-16T15:31:37Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"63af942518340e5a756c95709addc1847e8c0615a5e6dda7c01c81789c4eaa6c","abstract_canon_sha256":"71b1dad30ade25818d3000285547ca8eff6bed42f4850dbf1feeb6eaa3f5ab98"},"schema_version":"1.0"},"canonical_sha256":"6937eec4a1e4f00460ffd0720f0274ac68e259cdfc4bf58f5967cce80b0db6d2","source":{"kind":"arxiv","id":"1904.07763","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.07763","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.07763v1","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.07763","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"NE365RFB4TYA","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"NE365RFB4TYAIYH7","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"NE365RFB","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:NE365RFB4TYAIYH72BZA6ATUVR","target":"record","payload":{"canonical_record":{"source":{"id":"1904.07763","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2019-04-16T15:31:37Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"63af942518340e5a756c95709addc1847e8c0615a5e6dda7c01c81789c4eaa6c","abstract_canon_sha256":"71b1dad30ade25818d3000285547ca8eff6bed42f4850dbf1feeb6eaa3f5ab98"},"schema_version":"1.0"},"canonical_sha256":"6937eec4a1e4f00460ffd0720f0274ac68e259cdfc4bf58f5967cce80b0db6d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:24.295377Z","signature_b64":"Dd2EQr/WeXWc0VIBpy4Z8DyDgyElrEPJZmGy23pTuELSAaTKowUUCex9in6d1UScUM4D/UXtbz5mrajxQtuYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6937eec4a1e4f00460ffd0720f0274ac68e259cdfc4bf58f5967cce80b0db6d2","last_reissued_at":"2026-05-17T23:48:24.294759Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:24.294759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.07763","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":"2026-05-17T23:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fxbF4C2Qs78Dkxt26s+YjIlglvvkzz8PHkm0wcBPKUAsJ0fAOxaoi2+A2hMerOpw2X85XR0JYFAk0FlTzeB3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:01:09.388560Z"},"content_sha256":"86556dcf2f1c499746f48f7c9f280f8625174bbd3a1b75704560ae715ef38946","schema_version":"1.0","event_id":"sha256:86556dcf2f1c499746f48f7c9f280f8625174bbd3a1b75704560ae715ef38946"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:NE365RFB4TYAIYH72BZA6ATUVR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multidimensional Scaling: Infinite Metric Measure Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Lara Kassab","submitted_at":"2019-04-16T15:31:37Z","abstract_excerpt":"Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle $S^1$ into $\\mathbb{R}^m$ for all $m$, and to ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $\\mathbb{R}^m$. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07763","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":"2026-05-17T23:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lJSj1rjdpAAD3JicdR12qOfuaCPzwWpQR5ZWpStGktH1kD56nYNzRYMqh7VdYIPLLcML1NQ3I6OytabyX82rDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T09:01:09.388908Z"},"content_sha256":"f53b1a9f969e695451677356fb61018a5d7a9c5bbbcfbdc31af9742ad9ea20f2","schema_version":"1.0","event_id":"sha256:f53b1a9f969e695451677356fb61018a5d7a9c5bbbcfbdc31af9742ad9ea20f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NE365RFB4TYAIYH72BZA6ATUVR/bundle.json","state_url":"https://pith.science/pith/NE365RFB4TYAIYH72BZA6ATUVR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NE365RFB4TYAIYH72BZA6ATUVR/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":"2026-07-07T09:01:09Z","links":{"resolver":"https://pith.science/pith/NE365RFB4TYAIYH72BZA6ATUVR","bundle":"https://pith.science/pith/NE365RFB4TYAIYH72BZA6ATUVR/bundle.json","state":"https://pith.science/pith/NE365RFB4TYAIYH72BZA6ATUVR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NE365RFB4TYAIYH72BZA6ATUVR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:NE365RFB4TYAIYH72BZA6ATUVR","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":"71b1dad30ade25818d3000285547ca8eff6bed42f4850dbf1feeb6eaa3f5ab98","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2019-04-16T15:31:37Z","title_canon_sha256":"63af942518340e5a756c95709addc1847e8c0615a5e6dda7c01c81789c4eaa6c"},"schema_version":"1.0","source":{"id":"1904.07763","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.07763","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.07763v1","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.07763","created_at":"2026-05-17T23:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"NE365RFB4TYA","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"NE365RFB4TYAIYH7","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"NE365RFB","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:f53b1a9f969e695451677356fb61018a5d7a9c5bbbcfbdc31af9742ad9ea20f2","target":"graph","created_at":"2026-05-17T23:48:24Z","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":"Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle $S^1$ into $\\mathbb{R}^m$ for all $m$, and to ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $\\mathbb{R}^m$. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measu","authors_text":"Lara Kassab","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2019-04-16T15:31:37Z","title":"Multidimensional Scaling: Infinite Metric Measure Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07763","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":"sha256:86556dcf2f1c499746f48f7c9f280f8625174bbd3a1b75704560ae715ef38946","target":"record","created_at":"2026-05-17T23:48:24Z","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":"71b1dad30ade25818d3000285547ca8eff6bed42f4850dbf1feeb6eaa3f5ab98","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2019-04-16T15:31:37Z","title_canon_sha256":"63af942518340e5a756c95709addc1847e8c0615a5e6dda7c01c81789c4eaa6c"},"schema_version":"1.0","source":{"id":"1904.07763","kind":"arxiv","version":1}},"canonical_sha256":"6937eec4a1e4f00460ffd0720f0274ac68e259cdfc4bf58f5967cce80b0db6d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6937eec4a1e4f00460ffd0720f0274ac68e259cdfc4bf58f5967cce80b0db6d2","first_computed_at":"2026-05-17T23:48:24.294759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:24.294759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Dd2EQr/WeXWc0VIBpy4Z8DyDgyElrEPJZmGy23pTuELSAaTKowUUCex9in6d1UScUM4D/UXtbz5mrajxQtuYBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:24.295377Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.07763","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86556dcf2f1c499746f48f7c9f280f8625174bbd3a1b75704560ae715ef38946","sha256:f53b1a9f969e695451677356fb61018a5d7a9c5bbbcfbdc31af9742ad9ea20f2"],"state_sha256":"fbe01d999da06578d34dd1a49d64ec4aaf52f794e7c6f47c36461f6355568f66"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4QdGgGNK2GccRndn5XVuE+1xmHccUgFgv4/GdTRuBnkgDNqZ5SH+pDowdexaZeK40879+17Y3dLzTyVvnxRDBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T09:01:09.390925Z","bundle_sha256":"fa1e73be0de0decd7963771afdf30bdba2cb61612afdb0b9cdd375711ef9258b"}}