{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5BBWBZXREUZM6Z5WUUWZYT2NDK","short_pith_number":"pith:5BBWBZXR","canonical_record":{"source":{"id":"1604.04054","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-04-14T07:23:56Z","cross_cats_sorted":[],"title_canon_sha256":"eac61d9b518925a0bdc1c694cab98a583408ca8ed28b1ce9a71cea4838197fe3","abstract_canon_sha256":"7aafc32d9cb6e9c54dd7827ee6801c2d3b0ad614d08a5e8c1980885f4934bc5e"},"schema_version":"1.0"},"canonical_sha256":"e84360e6f12532cf67b6a52d9c4f4d1a8ce387ad7506882bd5098ee52aa3ff8c","source":{"kind":"arxiv","id":"1604.04054","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.04054","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"arxiv_version","alias_value":"1604.04054v1","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.04054","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"pith_short_12","alias_value":"5BBWBZXREUZM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"5BBWBZXREUZM6Z5W","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"5BBWBZXR","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5BBWBZXREUZM6Z5WUUWZYT2NDK","target":"record","payload":{"canonical_record":{"source":{"id":"1604.04054","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-04-14T07:23:56Z","cross_cats_sorted":[],"title_canon_sha256":"eac61d9b518925a0bdc1c694cab98a583408ca8ed28b1ce9a71cea4838197fe3","abstract_canon_sha256":"7aafc32d9cb6e9c54dd7827ee6801c2d3b0ad614d08a5e8c1980885f4934bc5e"},"schema_version":"1.0"},"canonical_sha256":"e84360e6f12532cf67b6a52d9c4f4d1a8ce387ad7506882bd5098ee52aa3ff8c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:06.830299Z","signature_b64":"tLFpN/dHzqseSTuvalXA+Zo8dfwyBgfjyRvgCsdwlpvxvpBYx33umey3EhCRnVFOYU2NDgVwtuGC678KUY/jAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e84360e6f12532cf67b6a52d9c4f4d1a8ce387ad7506882bd5098ee52aa3ff8c","last_reissued_at":"2026-05-18T01:17:06.829563Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:06.829563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.04054","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":"2026-05-18T01:17:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+aHF0e96+FYc4oESzQbde6T1EQt7obnMA82lYeSKBBBXszgt7ox043qCuKlrzntVWuxoFBT3E1Wfr+Pmi32iBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T02:46:38.924309Z"},"content_sha256":"470f01088c70e163c1c851e3a2d08d80d0e59b052bd4120cb2841e5423b42dbc","schema_version":"1.0","event_id":"sha256:470f01088c70e163c1c851e3a2d08d80d0e59b052bd4120cb2841e5423b42dbc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5BBWBZXREUZM6Z5WUUWZYT2NDK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Rates For Regularization Of Statistical Inverse Learning Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ML","authors_text":"Gilles Blanchard, Nicole M\\\"ucke","submitted_at":"2016-04-14T07:23:56Z","abstract_excerpt":"We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of $Af$) and the inverse (estimation of $f$) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations $n$ grows large) for a large class of spectral regularization methods over regularity classe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04054","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":"2026-05-18T01:17:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/osu6Mmm3YDIOXgX67CoJ55+jQ4JNa/rZ6mlOV1oDp7BISdIzkef8QixYbKhJEGzZ48B3WE9kBRRUn62KehxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T02:46:38.924681Z"},"content_sha256":"8e66225fa54f71aa1897ad33d7fe0994dd3020673471f8950f84b2c7740b1276","schema_version":"1.0","event_id":"sha256:8e66225fa54f71aa1897ad33d7fe0994dd3020673471f8950f84b2c7740b1276"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/bundle.json","state_url":"https://pith.science/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/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":"2026-06-23T02:46:38Z","links":{"resolver":"https://pith.science/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK","bundle":"https://pith.science/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/bundle.json","state":"https://pith.science/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5BBWBZXREUZM6Z5WUUWZYT2NDK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5BBWBZXREUZM6Z5WUUWZYT2NDK","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":"7aafc32d9cb6e9c54dd7827ee6801c2d3b0ad614d08a5e8c1980885f4934bc5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-04-14T07:23:56Z","title_canon_sha256":"eac61d9b518925a0bdc1c694cab98a583408ca8ed28b1ce9a71cea4838197fe3"},"schema_version":"1.0","source":{"id":"1604.04054","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.04054","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"arxiv_version","alias_value":"1604.04054v1","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.04054","created_at":"2026-05-18T01:17:06Z"},{"alias_kind":"pith_short_12","alias_value":"5BBWBZXREUZM","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"5BBWBZXREUZM6Z5W","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"5BBWBZXR","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:8e66225fa54f71aa1897ad33d7fe0994dd3020673471f8950f84b2c7740b1276","target":"graph","created_at":"2026-05-18T01:17:06Z","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":"We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of $Af$) and the inverse (estimation of $f$) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations $n$ grows large) for a large class of spectral regularization methods over regularity classe","authors_text":"Gilles Blanchard, Nicole M\\\"ucke","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-04-14T07:23:56Z","title":"Optimal Rates For Regularization Of Statistical Inverse Learning Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04054","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":"sha256:470f01088c70e163c1c851e3a2d08d80d0e59b052bd4120cb2841e5423b42dbc","target":"record","created_at":"2026-05-18T01:17:06Z","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":"7aafc32d9cb6e9c54dd7827ee6801c2d3b0ad614d08a5e8c1980885f4934bc5e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-04-14T07:23:56Z","title_canon_sha256":"eac61d9b518925a0bdc1c694cab98a583408ca8ed28b1ce9a71cea4838197fe3"},"schema_version":"1.0","source":{"id":"1604.04054","kind":"arxiv","version":1}},"canonical_sha256":"e84360e6f12532cf67b6a52d9c4f4d1a8ce387ad7506882bd5098ee52aa3ff8c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e84360e6f12532cf67b6a52d9c4f4d1a8ce387ad7506882bd5098ee52aa3ff8c","first_computed_at":"2026-05-18T01:17:06.829563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:06.829563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tLFpN/dHzqseSTuvalXA+Zo8dfwyBgfjyRvgCsdwlpvxvpBYx33umey3EhCRnVFOYU2NDgVwtuGC678KUY/jAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:06.830299Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.04054","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:470f01088c70e163c1c851e3a2d08d80d0e59b052bd4120cb2841e5423b42dbc","sha256:8e66225fa54f71aa1897ad33d7fe0994dd3020673471f8950f84b2c7740b1276"],"state_sha256":"0fd94070525d5c2c704c90e9bb6813d9d6a983c1757dac2234b70322eeaeba52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4UUOX4lvG3dHBUp5Ps66A4Va8Euty1EbU6mYyGQ/h7dA8ffv2AU+ymXQa9p0Z8/JhpqT3/bhmlL7dBHsTW7qCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T02:46:38.926643Z","bundle_sha256":"0e938a6561f51832c98e838b3a88df609423670762ae422070ae0398bb7532dc"}}