{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:SXCF3O3VI33OKDZZOBF4FDOZSS","short_pith_number":"pith:SXCF3O3V","canonical_record":{"source":{"id":"2509.08547","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-09-10T12:52:47Z","cross_cats_sorted":["math.AP","math.FA","math.PR"],"title_canon_sha256":"a1b9a828ebefa8544238e4fec3466dac1255aeff2f8de074202808fc1d5a92ad","abstract_canon_sha256":"52cd4fd01ebbfe4c0344930a6816f4e97f503d9f40ecd8ab77b27a5da056a59c"},"schema_version":"1.0"},"canonical_sha256":"95c45dbb7546f6e50f39704bc28dd9948123dc80247849a4c3dc267f3c707fd6","source":{"kind":"arxiv","id":"2509.08547","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08547","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08547v3","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08547","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"SXCF3O3VI33O","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_16","alias_value":"SXCF3O3VI33OKDZZ","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_8","alias_value":"SXCF3O3V","created_at":"2026-05-20T02:05:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:SXCF3O3VI33OKDZZOBF4FDOZSS","target":"record","payload":{"canonical_record":{"source":{"id":"2509.08547","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-09-10T12:52:47Z","cross_cats_sorted":["math.AP","math.FA","math.PR"],"title_canon_sha256":"a1b9a828ebefa8544238e4fec3466dac1255aeff2f8de074202808fc1d5a92ad","abstract_canon_sha256":"52cd4fd01ebbfe4c0344930a6816f4e97f503d9f40ecd8ab77b27a5da056a59c"},"schema_version":"1.0"},"canonical_sha256":"95c45dbb7546f6e50f39704bc28dd9948123dc80247849a4c3dc267f3c707fd6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T02:05:35.658510Z","signature_b64":"zrKG/0SFafYZDEtiJLxEGVmhnvLJUWgcMi5FiDvZGHQLyiWnGOve7Yw7YmSVkP/aMLyUhjE9ALbv22LMeF2zBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95c45dbb7546f6e50f39704bc28dd9948123dc80247849a4c3dc267f3c707fd6","last_reissued_at":"2026-05-20T02:05:35.657628Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T02:05:35.657628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.08547","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T02:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ZP4li6YQfMTtret6QkBqET35fzJHH9vR0U6puRsqj59pGRbfuY0lz/eJhZp9swdhI5U6cLQulSxFUUlsAQaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T06:06:01.733073Z"},"content_sha256":"01f12fcfd2df3147746e1be4549b4004c378596ae4447d0c8d15020ff7339ae6","schema_version":"1.0","event_id":"sha256:01f12fcfd2df3147746e1be4549b4004c378596ae4447d0c8d15020ff7339ae6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:SXCF3O3VI33OKDZZOBF4FDOZSS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA","math.PR"],"primary_cat":"math.OC","authors_text":"Alberto Gonz\\'alez-Sanz, Andr\\'es Riveros Valdevenito, Marcel Nutz","submitted_at":"2025-09-10T12:52:47Z","abstract_excerpt":"In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Quadratic regularization penalizes transport couplings by the squared $L^2$ norm of their density, or equivalently by the $\\chi^2$ divergence. While a number of computational approaches have been shown to work in practice, the dual problem is not strongly convex and theoretical convergence results are scarce. We focus on the dual gradient descent algorithm in a continuous setting and establish linear convergence in $L^2$, that is, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08547","kind":"arxiv","version":3},"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/2509.08547/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":"2026-05-20T02:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ahZ5bmRWH91pXXnokoH7X3ptZvY71epl3DwQv7qv3Ike7tLVR4hxdirojanhK/v4N5xGRY4vphDo22EIZqaECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T06:06:01.733784Z"},"content_sha256":"62d801803ab6f409102e199951cbe23057008f634f9770f339d29b86691a6b36","schema_version":"1.0","event_id":"sha256:62d801803ab6f409102e199951cbe23057008f634f9770f339d29b86691a6b36"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/bundle.json","state_url":"https://pith.science/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/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":"2026-05-31T06:06:01Z","links":{"resolver":"https://pith.science/pith/SXCF3O3VI33OKDZZOBF4FDOZSS","bundle":"https://pith.science/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/bundle.json","state":"https://pith.science/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SXCF3O3VI33OKDZZOBF4FDOZSS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:SXCF3O3VI33OKDZZOBF4FDOZSS","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":"52cd4fd01ebbfe4c0344930a6816f4e97f503d9f40ecd8ab77b27a5da056a59c","cross_cats_sorted":["math.AP","math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-09-10T12:52:47Z","title_canon_sha256":"a1b9a828ebefa8544238e4fec3466dac1255aeff2f8de074202808fc1d5a92ad"},"schema_version":"1.0","source":{"id":"2509.08547","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08547","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08547v3","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08547","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"SXCF3O3VI33O","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_16","alias_value":"SXCF3O3VI33OKDZZ","created_at":"2026-05-20T02:05:35Z"},{"alias_kind":"pith_short_8","alias_value":"SXCF3O3V","created_at":"2026-05-20T02:05:35Z"}],"graph_snapshots":[{"event_id":"sha256:62d801803ab6f409102e199951cbe23057008f634f9770f339d29b86691a6b36","target":"graph","created_at":"2026-05-20T02:05:35Z","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/2509.08547/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Quadratic regularization penalizes transport couplings by the squared $L^2$ norm of their density, or equivalently by the $\\chi^2$ divergence. While a number of computational approaches have been shown to work in practice, the dual problem is not strongly convex and theoretical convergence results are scarce. We focus on the dual gradient descent algorithm in a continuous setting and establish linear convergence in $L^2$, that is, the","authors_text":"Alberto Gonz\\'alez-Sanz, Andr\\'es Riveros Valdevenito, Marcel Nutz","cross_cats":["math.AP","math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-09-10T12:52:47Z","title":"Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08547","kind":"arxiv","version":3},"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":"sha256:01f12fcfd2df3147746e1be4549b4004c378596ae4447d0c8d15020ff7339ae6","target":"record","created_at":"2026-05-20T02:05:35Z","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":"52cd4fd01ebbfe4c0344930a6816f4e97f503d9f40ecd8ab77b27a5da056a59c","cross_cats_sorted":["math.AP","math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2025-09-10T12:52:47Z","title_canon_sha256":"a1b9a828ebefa8544238e4fec3466dac1255aeff2f8de074202808fc1d5a92ad"},"schema_version":"1.0","source":{"id":"2509.08547","kind":"arxiv","version":3}},"canonical_sha256":"95c45dbb7546f6e50f39704bc28dd9948123dc80247849a4c3dc267f3c707fd6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95c45dbb7546f6e50f39704bc28dd9948123dc80247849a4c3dc267f3c707fd6","first_computed_at":"2026-05-20T02:05:35.657628Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T02:05:35.657628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zrKG/0SFafYZDEtiJLxEGVmhnvLJUWgcMi5FiDvZGHQLyiWnGOve7Yw7YmSVkP/aMLyUhjE9ALbv22LMeF2zBQ==","signature_status":"signed_v1","signed_at":"2026-05-20T02:05:35.658510Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.08547","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01f12fcfd2df3147746e1be4549b4004c378596ae4447d0c8d15020ff7339ae6","sha256:62d801803ab6f409102e199951cbe23057008f634f9770f339d29b86691a6b36"],"state_sha256":"2bdbc79b1cca95819a0844734699495721e96c998da8bd7e14b716c838ff01b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qMtJwDE5mxScW86U3LabadhLyr0qerdLsCnyqC5hIvZpikkdIOrP+M8PjIS6bQOMuV6h2keb9PcpqnTW66b+Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T06:06:01.737311Z","bundle_sha256":"1dbce45a9d4f9ea8b6e95bcaca7e474b6c510e2e5b59ba8691cdd74797cd6232"}}