{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:WAPQOS36GXM7E4OJFTF75UQLUF","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":"9bd94319a2e6e79b1942bd0c2111ea8d47d470bf04392544e31066d3912d92db","cross_cats_sorted":["cs.IT","math.FA","math.IT","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-04T20:20:59Z","title_canon_sha256":"4c129cb57b1d62c8c459149fe1ab75908d0063a3af008145c15b42205c4aae01"},"schema_version":"1.0","source":{"id":"2506.04425","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.04425","created_at":"2026-07-05T11:16:14Z"},{"alias_kind":"arxiv_version","alias_value":"2506.04425v1","created_at":"2026-07-05T11:16:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.04425","created_at":"2026-07-05T11:16:14Z"},{"alias_kind":"pith_short_12","alias_value":"WAPQOS36GXM7","created_at":"2026-07-05T11:16:14Z"},{"alias_kind":"pith_short_16","alias_value":"WAPQOS36GXM7E4OJ","created_at":"2026-07-05T11:16:14Z"},{"alias_kind":"pith_short_8","alias_value":"WAPQOS36","created_at":"2026-07-05T11:16:14Z"}],"graph_snapshots":[{"event_id":"sha256:67b20edaac2c59a9b2e931047e850e53c575a4860debe5b6b484e0667ed5c7ea","target":"graph","created_at":"2026-07-05T11:16:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.04425/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.","authors_text":"Ben Blum-Smith, Brantley Vose, Dustin G. Mixon, Harm Derksen, Yousef Qaddura","cross_cats":["cs.IT","math.FA","math.IT","math.RT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-04T20:20:59Z","title":"Estimating the Euclidean distortion of an orbit space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04425","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":"sha256:691ff94d8e2e0169b7ce50ac76b27355fb69e76eaf1443ffa3345d32aa8e160c","target":"record","created_at":"2026-07-05T11:16:14Z","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":"9bd94319a2e6e79b1942bd0c2111ea8d47d470bf04392544e31066d3912d92db","cross_cats_sorted":["cs.IT","math.FA","math.IT","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-06-04T20:20:59Z","title_canon_sha256":"4c129cb57b1d62c8c459149fe1ab75908d0063a3af008145c15b42205c4aae01"},"schema_version":"1.0","source":{"id":"2506.04425","kind":"arxiv","version":1}},"canonical_sha256":"b01f074b7e35d9f271c92ccbfed20ba17b2a22bf1e85081f5ca165267ca4fa36","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b01f074b7e35d9f271c92ccbfed20ba17b2a22bf1e85081f5ca165267ca4fa36","first_computed_at":"2026-07-05T11:16:14.151524Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:16:14.151524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CslNyyWAvQke6+9xSRIK7JLM6f+lYRqZIbSdU6OcMvmpTkjNstSbHiU/78bcOhzVIiJkIODbDUVqGQu6Rhd7Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:16:14.151961Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.04425","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:691ff94d8e2e0169b7ce50ac76b27355fb69e76eaf1443ffa3345d32aa8e160c","sha256:67b20edaac2c59a9b2e931047e850e53c575a4860debe5b6b484e0667ed5c7ea"],"state_sha256":"4e09c53b909c7f250f252a61a3b6f22aa1fc6da368889577ea68bf7f5e29ce20"}