{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7M2KRQTDX3QZ3YX7B2KQEJSZG6","short_pith_number":"pith:7M2KRQTD","canonical_record":{"source":{"id":"1705.03437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-09T17:24:07Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"75c026d911a38a6f10f0785037bf490e9cdd94c0193a758e620649d70d1a599e","abstract_canon_sha256":"44e325d5d0a8cf3f416581ecfd1ccb6de5b2d66ca1419203ed358ff6f9b6c0e7"},"schema_version":"1.0"},"canonical_sha256":"fb34a8c263bee19de2ff0e9502265937a2099522c523ed0a98e2942a742846f7","source":{"kind":"arxiv","id":"1705.03437","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.03437","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"arxiv_version","alias_value":"1705.03437v1","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.03437","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"pith_short_12","alias_value":"7M2KRQTDX3QZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7M2KRQTDX3QZ3YX7","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7M2KRQTD","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7M2KRQTDX3QZ3YX7B2KQEJSZG6","target":"record","payload":{"canonical_record":{"source":{"id":"1705.03437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-09T17:24:07Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"75c026d911a38a6f10f0785037bf490e9cdd94c0193a758e620649d70d1a599e","abstract_canon_sha256":"44e325d5d0a8cf3f416581ecfd1ccb6de5b2d66ca1419203ed358ff6f9b6c0e7"},"schema_version":"1.0"},"canonical_sha256":"fb34a8c263bee19de2ff0e9502265937a2099522c523ed0a98e2942a742846f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:47.846978Z","signature_b64":"Da7WA/r+PaF4yvEVDNYcCyyInHMAcJZAu8bjwnMDobBjPyH8LV4P/qEo32+08WwxEyl4wPsLpCcb7LnbOBYpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb34a8c263bee19de2ff0e9502265937a2099522c523ed0a98e2942a742846f7","last_reissued_at":"2026-05-18T00:44:47.846514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:47.846514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.03437","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":"2026-05-18T00:44:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Krc/q5Ix0iyDTf8pVcFEJn5qzwc14AwolYWGcJuOkGp07KAWiLuLcmIRuh5PgOjpWEqauYgs8mqAJRFYnDqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:37:48.854289Z"},"content_sha256":"d3b7fae856e41e3f06f3cd6dd629c379899f427a608234457d80f65a49105718","schema_version":"1.0","event_id":"sha256:d3b7fae856e41e3f06f3cd6dd629c379899f427a608234457d80f65a49105718"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7M2KRQTDX3QZ3YX7B2KQEJSZG6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal properties of the canonical tight probabilistic frame","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CA","authors_text":"Desai Cheng, Kasso A. Okoudjou","submitted_at":"2017-05-09T17:24:07Z","abstract_excerpt":"A probabilistic frame is a Borel probability measure with finite second moment whose support spans $\\mathbb{R}^d$. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in $\\mathbb{R}^d$. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the $2-$Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabili"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03437","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":"2026-05-18T00:44:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FB1oUAsKiTrwCO7JIn80y+PMiQ8kjcnloPA/pKGY89lq4DqFkzfVbsaqogBtDT2STWELDgX/fnp565X/lK4SAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:37:48.854658Z"},"content_sha256":"ea9f8e09abd685de2e2fdc00ce09976b05d8a461145e672f056d95b43a891599","schema_version":"1.0","event_id":"sha256:ea9f8e09abd685de2e2fdc00ce09976b05d8a461145e672f056d95b43a891599"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/bundle.json","state_url":"https://pith.science/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/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":"2026-06-27T17:37:48Z","links":{"resolver":"https://pith.science/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6","bundle":"https://pith.science/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/bundle.json","state":"https://pith.science/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7M2KRQTDX3QZ3YX7B2KQEJSZG6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7M2KRQTDX3QZ3YX7B2KQEJSZG6","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":"44e325d5d0a8cf3f416581ecfd1ccb6de5b2d66ca1419203ed358ff6f9b6c0e7","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-09T17:24:07Z","title_canon_sha256":"75c026d911a38a6f10f0785037bf490e9cdd94c0193a758e620649d70d1a599e"},"schema_version":"1.0","source":{"id":"1705.03437","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.03437","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"arxiv_version","alias_value":"1705.03437v1","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.03437","created_at":"2026-05-18T00:44:47Z"},{"alias_kind":"pith_short_12","alias_value":"7M2KRQTDX3QZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7M2KRQTDX3QZ3YX7","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7M2KRQTD","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:ea9f8e09abd685de2e2fdc00ce09976b05d8a461145e672f056d95b43a891599","target":"graph","created_at":"2026-05-18T00:44:47Z","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":"A probabilistic frame is a Borel probability measure with finite second moment whose support spans $\\mathbb{R}^d$. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in $\\mathbb{R}^d$. Each probabilistic frame is canonically associated to a Parseval probabilistic frame. In this paper, we show that this canonical Parseval probabilistic frame is the closest Parseval probabilistic frame to a given probabilistic frame in the $2-$Wasserstein distance. Our proof is based on two main ingredients. On the one hand, we show that a probabili","authors_text":"Desai Cheng, Kasso A. Okoudjou","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-09T17:24:07Z","title":"Optimal properties of the canonical tight probabilistic frame"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03437","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:d3b7fae856e41e3f06f3cd6dd629c379899f427a608234457d80f65a49105718","target":"record","created_at":"2026-05-18T00:44:47Z","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":"44e325d5d0a8cf3f416581ecfd1ccb6de5b2d66ca1419203ed358ff6f9b6c0e7","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-05-09T17:24:07Z","title_canon_sha256":"75c026d911a38a6f10f0785037bf490e9cdd94c0193a758e620649d70d1a599e"},"schema_version":"1.0","source":{"id":"1705.03437","kind":"arxiv","version":1}},"canonical_sha256":"fb34a8c263bee19de2ff0e9502265937a2099522c523ed0a98e2942a742846f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb34a8c263bee19de2ff0e9502265937a2099522c523ed0a98e2942a742846f7","first_computed_at":"2026-05-18T00:44:47.846514Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:47.846514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Da7WA/r+PaF4yvEVDNYcCyyInHMAcJZAu8bjwnMDobBjPyH8LV4P/qEo32+08WwxEyl4wPsLpCcb7LnbOBYpCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:47.846978Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.03437","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3b7fae856e41e3f06f3cd6dd629c379899f427a608234457d80f65a49105718","sha256:ea9f8e09abd685de2e2fdc00ce09976b05d8a461145e672f056d95b43a891599"],"state_sha256":"f452534d3cf816fac46de099dc64a025244062fdc7e5312390d98aecd1ef388c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WF93qXsdrRBEXeAaPwe71pP//By3fO2aTRNJn6FPJB9f8jM6cu70tqvYOkA7gVFvI04DXguISudYCynfSUZwDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T17:37:48.856557Z","bundle_sha256":"1d03f62b71944ae0047f85f7d699f19fec98461ac9dddc45a413c37f3a28f017"}}