{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:37F5PFCZKBKXWYD7FIKI6DMASW","short_pith_number":"pith:37F5PFCZ","canonical_record":{"source":{"id":"1602.04361","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-13T17:53:48Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"75b86f73a7719cf8397cbb73946efb0af1004cc321fac4c1e8f3dfac9f4fbc3b","abstract_canon_sha256":"06397700d9186d9537d5954296cbf530c3d98c9451fc54f9b1ac7c805bb0866c"},"schema_version":"1.0"},"canonical_sha256":"dfcbd7945950557b607f2a148f0d8095b337abd5cbd78b9a2a7a26ca734e1f49","source":{"kind":"arxiv","id":"1602.04361","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04361","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04361v2","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04361","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"37F5PFCZKBKX","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"37F5PFCZKBKXWYD7","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"37F5PFCZ","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:37F5PFCZKBKXWYD7FIKI6DMASW","target":"record","payload":{"canonical_record":{"source":{"id":"1602.04361","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-13T17:53:48Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"75b86f73a7719cf8397cbb73946efb0af1004cc321fac4c1e8f3dfac9f4fbc3b","abstract_canon_sha256":"06397700d9186d9537d5954296cbf530c3d98c9451fc54f9b1ac7c805bb0866c"},"schema_version":"1.0"},"canonical_sha256":"dfcbd7945950557b607f2a148f0d8095b337abd5cbd78b9a2a7a26ca734e1f49","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:07.216484Z","signature_b64":"gtVMtRJNOs2+tHeMa1uMhaVJ7BUP017YsjT7BlvmSPkwu9FmjaY2JGzF9GzzRigQyT/tVuU+wmmAF1freWacAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfcbd7945950557b607f2a148f0d8095b337abd5cbd78b9a2a7a26ca734e1f49","last_reissued_at":"2026-05-18T00:39:07.215841Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:07.215841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.04361","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":"2026-05-18T00:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ewPH+bIIAvQlwqOuMXEh3voEm+C4H5e1yW/Bg2s5kv/nq30zklElKeUZQ2PNkfx6fa9S1lQ3pHIwNMN8vfRLCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:07:25.498117Z"},"content_sha256":"49b254ef8000389089dd8d7b8b0e54cd97a898dba22c12d858d654bd6372cc2c","schema_version":"1.0","event_id":"sha256:49b254ef8000389089dd8d7b8b0e54cd97a898dba22c12d858d654bd6372cc2c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:37F5PFCZKBKXWYD7FIKI6DMASW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimax Estimation of Kernel Mean Embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Bharath Sriperumbudur, Ilya Tolstikhin, Krikamol Muandet","submitted_at":"2016-02-13T17:53:48Z","abstract_excerpt":"In this paper, we study the minimax estimation of the Bochner integral $$\\mu_k(P):=\\int_{\\mathcal{X}} k(\\cdot,x)\\,dP(x),$$ also called as the kernel mean embedding, based on random samples drawn i.i.d.~from $P$, where $k:\\mathcal{X}\\times\\mathcal{X}\\rightarrow\\mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\\hat{\\theta}_n$ of $\\mu_k(P)$ are studied in the literature wherein all of them satisfy $\\bigl\\| \\hat{\\theta}_n-\\mu_k(P)\\bigr\\|_{\\mathcal{H}_k}=O_P(n^{-1/2})$ with $\\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04361","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":""},"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":"2026-05-18T00:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LgNBU8YAxF3PsSvR12XDOYEZTWYzlikqY7vg5OTkNip6MxAzzz++y9tVA7uALaiD9xjmmlGQ2WJHQpY56UHRCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:07:25.498493Z"},"content_sha256":"1fc594d6fb4052a128bf253315c096e105e7021a153c4f4a7124e4e4c7d3bab0","schema_version":"1.0","event_id":"sha256:1fc594d6fb4052a128bf253315c096e105e7021a153c4f4a7124e4e4c7d3bab0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/37F5PFCZKBKXWYD7FIKI6DMASW/bundle.json","state_url":"https://pith.science/pith/37F5PFCZKBKXWYD7FIKI6DMASW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/37F5PFCZKBKXWYD7FIKI6DMASW/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":"2026-05-26T09:07:25Z","links":{"resolver":"https://pith.science/pith/37F5PFCZKBKXWYD7FIKI6DMASW","bundle":"https://pith.science/pith/37F5PFCZKBKXWYD7FIKI6DMASW/bundle.json","state":"https://pith.science/pith/37F5PFCZKBKXWYD7FIKI6DMASW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/37F5PFCZKBKXWYD7FIKI6DMASW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:37F5PFCZKBKXWYD7FIKI6DMASW","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":"06397700d9186d9537d5954296cbf530c3d98c9451fc54f9b1ac7c805bb0866c","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-13T17:53:48Z","title_canon_sha256":"75b86f73a7719cf8397cbb73946efb0af1004cc321fac4c1e8f3dfac9f4fbc3b"},"schema_version":"1.0","source":{"id":"1602.04361","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04361","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04361v2","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04361","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"37F5PFCZKBKX","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"37F5PFCZKBKXWYD7","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"37F5PFCZ","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:1fc594d6fb4052a128bf253315c096e105e7021a153c4f4a7124e4e4c7d3bab0","target":"graph","created_at":"2026-05-18T00:39:07Z","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":"In this paper, we study the minimax estimation of the Bochner integral $$\\mu_k(P):=\\int_{\\mathcal{X}} k(\\cdot,x)\\,dP(x),$$ also called as the kernel mean embedding, based on random samples drawn i.i.d.~from $P$, where $k:\\mathcal{X}\\times\\mathcal{X}\\rightarrow\\mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\\hat{\\theta}_n$ of $\\mu_k(P)$ are studied in the literature wherein all of them satisfy $\\bigl\\| \\hat{\\theta}_n-\\mu_k(P)\\bigr\\|_{\\mathcal{H}_k}=O_P(n^{-1/2})$ with $\\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. T","authors_text":"Bharath Sriperumbudur, Ilya Tolstikhin, Krikamol Muandet","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-13T17:53:48Z","title":"Minimax Estimation of Kernel Mean Embeddings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04361","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":"sha256:49b254ef8000389089dd8d7b8b0e54cd97a898dba22c12d858d654bd6372cc2c","target":"record","created_at":"2026-05-18T00:39:07Z","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":"06397700d9186d9537d5954296cbf530c3d98c9451fc54f9b1ac7c805bb0866c","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2016-02-13T17:53:48Z","title_canon_sha256":"75b86f73a7719cf8397cbb73946efb0af1004cc321fac4c1e8f3dfac9f4fbc3b"},"schema_version":"1.0","source":{"id":"1602.04361","kind":"arxiv","version":2}},"canonical_sha256":"dfcbd7945950557b607f2a148f0d8095b337abd5cbd78b9a2a7a26ca734e1f49","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfcbd7945950557b607f2a148f0d8095b337abd5cbd78b9a2a7a26ca734e1f49","first_computed_at":"2026-05-18T00:39:07.215841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:07.215841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gtVMtRJNOs2+tHeMa1uMhaVJ7BUP017YsjT7BlvmSPkwu9FmjaY2JGzF9GzzRigQyT/tVuU+wmmAF1freWacAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:07.216484Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.04361","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49b254ef8000389089dd8d7b8b0e54cd97a898dba22c12d858d654bd6372cc2c","sha256:1fc594d6fb4052a128bf253315c096e105e7021a153c4f4a7124e4e4c7d3bab0"],"state_sha256":"f64b6410f82b877b0de26329e07b743df130abca4ce50998e06f663b86ea4ac1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9w263Cjy18IH96xDPqQLdZSP1voGAg31lQoMpSKZ8ViVKo0HH5QPZcGvF+s6PiNzhg4bwmYZKCAgHQRuvQTeCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T09:07:25.502161Z","bundle_sha256":"baaefbf164adc57240bc54b500505518a92a63c6307e51945a3df3a7a3e386d3"}}