{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:DBFWZROY5MWZG6COPMWL4W7U6V","short_pith_number":"pith:DBFWZROY","canonical_record":{"source":{"id":"2512.09678","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-12-10T14:25:45Z","cross_cats_sorted":["cs.AI","cs.LG"],"title_canon_sha256":"68a57a3086f1574151bcb9b3b608bb2b74ec4e984ef9f03ec061c86e99c818f6","abstract_canon_sha256":"2ca3a3398946a01c8cad33f9060da04f5621794b8f33b5c5aa5a847968406bec"},"schema_version":"1.0"},"canonical_sha256":"184b6cc5d8eb2d93784e7b2cbe5bf4f55576a6db0eb44e4c5a9b1c2966ae175c","source":{"kind":"arxiv","id":"2512.09678","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2512.09678","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"arxiv_version","alias_value":"2512.09678v2","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.09678","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_12","alias_value":"DBFWZROY5MWZ","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_16","alias_value":"DBFWZROY5MWZG6CO","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_8","alias_value":"DBFWZROY","created_at":"2026-06-23T03:13:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:DBFWZROY5MWZG6COPMWL4W7U6V","target":"record","payload":{"canonical_record":{"source":{"id":"2512.09678","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-12-10T14:25:45Z","cross_cats_sorted":["cs.AI","cs.LG"],"title_canon_sha256":"68a57a3086f1574151bcb9b3b608bb2b74ec4e984ef9f03ec061c86e99c818f6","abstract_canon_sha256":"2ca3a3398946a01c8cad33f9060da04f5621794b8f33b5c5aa5a847968406bec"},"schema_version":"1.0"},"canonical_sha256":"184b6cc5d8eb2d93784e7b2cbe5bf4f55576a6db0eb44e4c5a9b1c2966ae175c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:13:52.308228Z","signature_b64":"EX+lhCYxWa6AkNv7g/S8hxAGe9NC14MWkw7B6pcJteYB0jAedf6j+nzTHKgcWWIovjiIT/4Ub0gGN0siXBTuCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"184b6cc5d8eb2d93784e7b2cbe5bf4f55576a6db0eb44e4c5a9b1c2966ae175c","last_reissued_at":"2026-06-23T03:13:52.307746Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:13:52.307746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2512.09678","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-23T03:13:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KgaRQXJSpPMYmNXzDlhLzAV3q2I1DLGvuM1rDPoT6f3pTmiDPDb6PBFg6mcKuzYvNKIWzpfda8bMo2pZjOfOBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T09:39:44.222835Z"},"content_sha256":"b7364ed6468b92145e390343e9bfdd836a9049a7c304f27948db8949b6341fd4","schema_version":"1.0","event_id":"sha256:b7364ed6468b92145e390343e9bfdd836a9049a7c304f27948db8949b6341fd4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:DBFWZROY5MWZG6COPMWL4W7U6V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ky Fan Norms and Beyond: Dual Norms and Combinations for Matrix Optimization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.LG"],"primary_cat":"math.OC","authors_text":"Alexander Vinogradov, Alexey Kravatskiy, Daniil Merkulov, Ivan Kozyrev, Ivan Oseledets, Nikolai Kozlov","submitted_at":"2025-12-10T14:25:45Z","abstract_excerpt":"In this article, we explore the use of various matrix norms for optimizing functions of weight matrices, a crucial problem in deep learning. Moving beyond the spectral norm that underlies the Muon update, we leverage the duals of the Ky Fan norms to introduce the Fanion family of linear minimization oracle (LMO) algorithms, which are closely related to Muon, $\\nu$-SAM, and Dion. Staying inside the LMO, we construct the families of F-Fanions and S-Fanions, whose updates are convex combinations of the updates of Fanions and Normalized SGD or SignSGD, respectively. The most promising algorithms i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.09678","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.09678/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-23T03:13:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PYX2iDtakMEraeTZtT4TVJIuOaPvkv/aXQ5JKNGy6zZHl5Yr71NwjWqv39CWKAhC+uFgJpNfMw5ot08NWBnZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T09:39:44.223213Z"},"content_sha256":"85080f9ef6385024941a9c4c3637d0a4c53d19923b269efd979a1c1596c38f47","schema_version":"1.0","event_id":"sha256:85080f9ef6385024941a9c4c3637d0a4c53d19923b269efd979a1c1596c38f47"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DBFWZROY5MWZG6COPMWL4W7U6V/bundle.json","state_url":"https://pith.science/pith/DBFWZROY5MWZG6COPMWL4W7U6V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DBFWZROY5MWZG6COPMWL4W7U6V/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":"2026-06-28T09:39:44Z","links":{"resolver":"https://pith.science/pith/DBFWZROY5MWZG6COPMWL4W7U6V","bundle":"https://pith.science/pith/DBFWZROY5MWZG6COPMWL4W7U6V/bundle.json","state":"https://pith.science/pith/DBFWZROY5MWZG6COPMWL4W7U6V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DBFWZROY5MWZG6COPMWL4W7U6V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:DBFWZROY5MWZG6COPMWL4W7U6V","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":"2ca3a3398946a01c8cad33f9060da04f5621794b8f33b5c5aa5a847968406bec","cross_cats_sorted":["cs.AI","cs.LG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-12-10T14:25:45Z","title_canon_sha256":"68a57a3086f1574151bcb9b3b608bb2b74ec4e984ef9f03ec061c86e99c818f6"},"schema_version":"1.0","source":{"id":"2512.09678","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2512.09678","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"arxiv_version","alias_value":"2512.09678v2","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.09678","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_12","alias_value":"DBFWZROY5MWZ","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_16","alias_value":"DBFWZROY5MWZG6CO","created_at":"2026-06-23T03:13:52Z"},{"alias_kind":"pith_short_8","alias_value":"DBFWZROY","created_at":"2026-06-23T03:13:52Z"}],"graph_snapshots":[{"event_id":"sha256:85080f9ef6385024941a9c4c3637d0a4c53d19923b269efd979a1c1596c38f47","target":"graph","created_at":"2026-06-23T03:13:52Z","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/2512.09678/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we explore the use of various matrix norms for optimizing functions of weight matrices, a crucial problem in deep learning. Moving beyond the spectral norm that underlies the Muon update, we leverage the duals of the Ky Fan norms to introduce the Fanion family of linear minimization oracle (LMO) algorithms, which are closely related to Muon, $\\nu$-SAM, and Dion. Staying inside the LMO, we construct the families of F-Fanions and S-Fanions, whose updates are convex combinations of the updates of Fanions and Normalized SGD or SignSGD, respectively. The most promising algorithms i","authors_text":"Alexander Vinogradov, Alexey Kravatskiy, Daniil Merkulov, Ivan Kozyrev, Ivan Oseledets, Nikolai Kozlov","cross_cats":["cs.AI","cs.LG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-12-10T14:25:45Z","title":"Ky Fan Norms and Beyond: Dual Norms and Combinations for Matrix Optimization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.09678","kind":"arxiv","version":2},"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:b7364ed6468b92145e390343e9bfdd836a9049a7c304f27948db8949b6341fd4","target":"record","created_at":"2026-06-23T03:13:52Z","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":"2ca3a3398946a01c8cad33f9060da04f5621794b8f33b5c5aa5a847968406bec","cross_cats_sorted":["cs.AI","cs.LG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-12-10T14:25:45Z","title_canon_sha256":"68a57a3086f1574151bcb9b3b608bb2b74ec4e984ef9f03ec061c86e99c818f6"},"schema_version":"1.0","source":{"id":"2512.09678","kind":"arxiv","version":2}},"canonical_sha256":"184b6cc5d8eb2d93784e7b2cbe5bf4f55576a6db0eb44e4c5a9b1c2966ae175c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"184b6cc5d8eb2d93784e7b2cbe5bf4f55576a6db0eb44e4c5a9b1c2966ae175c","first_computed_at":"2026-06-23T03:13:52.307746Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T03:13:52.307746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EX+lhCYxWa6AkNv7g/S8hxAGe9NC14MWkw7B6pcJteYB0jAedf6j+nzTHKgcWWIovjiIT/4Ub0gGN0siXBTuCA==","signature_status":"signed_v1","signed_at":"2026-06-23T03:13:52.308228Z","signed_message":"canonical_sha256_bytes"},"source_id":"2512.09678","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b7364ed6468b92145e390343e9bfdd836a9049a7c304f27948db8949b6341fd4","sha256:85080f9ef6385024941a9c4c3637d0a4c53d19923b269efd979a1c1596c38f47"],"state_sha256":"b56cdc2aec81d877a394a26ef603d6fe67e4d05905615d2906fc28dba3fe88c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ikXj/5B1q31DWG/3ZyssDD/h/ZhmfRgVAQZXiLhX7DLQ5J6FfWH3UrTzPat3NvV2tMTAKYqAMFXWU2rGQl5iCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T09:39:44.225480Z","bundle_sha256":"6058b849f4700e6ac30d55fd99a3f77f064fcfc46c3a42b6791ed13656c4b27e"}}