{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:FHZYW6KVMLC4WJFFBUVMZVDEP3","short_pith_number":"pith:FHZYW6KV","canonical_record":{"source":{"id":"1802.10575","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-02-28T18:32:07Z","cross_cats_sorted":["cs.IT","cs.LG","math.IT","stat.TH"],"title_canon_sha256":"1c9308f141cfa8f54df5aeaf59a13c983bf78b0e6abbbf7a4107745aba10e3ba","abstract_canon_sha256":"8de42f7913e0b62786cc27f7574cb21c639fc53eb1a4995199bfb48f4ca9e21e"},"schema_version":"1.0"},"canonical_sha256":"29f38b795562c5cb24a50d2accd4647efcc5bc187b1acd079032098333426553","source":{"kind":"arxiv","id":"1802.10575","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.10575","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1802.10575v2","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.10575","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"FHZYW6KVMLC4","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"FHZYW6KVMLC4WJFF","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"FHZYW6KV","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:FHZYW6KVMLC4WJFFBUVMZVDEP3","target":"record","payload":{"canonical_record":{"source":{"id":"1802.10575","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-02-28T18:32:07Z","cross_cats_sorted":["cs.IT","cs.LG","math.IT","stat.TH"],"title_canon_sha256":"1c9308f141cfa8f54df5aeaf59a13c983bf78b0e6abbbf7a4107745aba10e3ba","abstract_canon_sha256":"8de42f7913e0b62786cc27f7574cb21c639fc53eb1a4995199bfb48f4ca9e21e"},"schema_version":"1.0"},"canonical_sha256":"29f38b795562c5cb24a50d2accd4647efcc5bc187b1acd079032098333426553","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:01.902274Z","signature_b64":"LS0Ha1TFNfUWy/jZ8aYSkUNeZCqSD2VPE+gKQvecFElD1UascYPaJ77LSs0necXNYu3ZMXOSk9DIKEyu4Py/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29f38b795562c5cb24a50d2accd4647efcc5bc187b1acd079032098333426553","last_reissued_at":"2026-05-17T23:59:01.901678Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:01.901678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.10575","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":"2026-05-17T23:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J0Rw6g4layZjl5h9c1C20at+AZsXJp2B2GUtSMQbZ4c3Cs9qhFzcEEioYBw5WipesqfoH8M8EsOBuUp3je+RDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:18:15.282428Z"},"content_sha256":"49c824fb8ab21d995ab29bac948005dc0bd1cff6b42b48bea1dd698eaa6a4741","schema_version":"1.0","event_id":"sha256:49c824fb8ab21d995ab29bac948005dc0bd1cff6b42b48bea1dd698eaa6a4741"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:FHZYW6KVMLC4WJFFBUVMZVDEP3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Alistair Stewart, Anastasios Sidiropoulos, Ilias Diakonikolas, Timothy Carpenter","submitted_at":"2018-02-28T18:32:07Z","abstract_excerpt":"We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\\mathbb{R}^d$, for all $d \\geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions.\n  In more detail, we prove that for any $d \\geq 1$ and $\\epsilon>0$, given $\\tilde{O}_d((1/\\epsilon)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\\mathbb{R}^d$, the MLE outputs a hypothesis $h$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10575","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":"2026-05-17T23:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fx9QpDfwj1xr28TKkaQHSBTYUCUAiPs/UzVidPu82xSievE8AFnqCXMZzBsHXsFO4noDIFnXcKgG8cwQBHtqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:18:15.282773Z"},"content_sha256":"5db31d8ab49e2ac41eb50777381eb06fa144f6b07d274ceebb890712dc938632","schema_version":"1.0","event_id":"sha256:5db31d8ab49e2ac41eb50777381eb06fa144f6b07d274ceebb890712dc938632"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/bundle.json","state_url":"https://pith.science/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/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":"2026-06-05T13:18:15Z","links":{"resolver":"https://pith.science/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3","bundle":"https://pith.science/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/bundle.json","state":"https://pith.science/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FHZYW6KVMLC4WJFFBUVMZVDEP3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:FHZYW6KVMLC4WJFFBUVMZVDEP3","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":"8de42f7913e0b62786cc27f7574cb21c639fc53eb1a4995199bfb48f4ca9e21e","cross_cats_sorted":["cs.IT","cs.LG","math.IT","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-02-28T18:32:07Z","title_canon_sha256":"1c9308f141cfa8f54df5aeaf59a13c983bf78b0e6abbbf7a4107745aba10e3ba"},"schema_version":"1.0","source":{"id":"1802.10575","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.10575","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1802.10575v2","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.10575","created_at":"2026-05-17T23:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"FHZYW6KVMLC4","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"FHZYW6KVMLC4WJFF","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"FHZYW6KV","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:5db31d8ab49e2ac41eb50777381eb06fa144f6b07d274ceebb890712dc938632","target":"graph","created_at":"2026-05-17T23:59:01Z","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 study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\\mathbb{R}^d$, for all $d \\geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions.\n  In more detail, we prove that for any $d \\geq 1$ and $\\epsilon>0$, given $\\tilde{O}_d((1/\\epsilon)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\\mathbb{R}^d$, the MLE outputs a hypothesis $h$ ","authors_text":"Alistair Stewart, Anastasios Sidiropoulos, Ilias Diakonikolas, Timothy Carpenter","cross_cats":["cs.IT","cs.LG","math.IT","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-02-28T18:32:07Z","title":"Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10575","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":"sha256:49c824fb8ab21d995ab29bac948005dc0bd1cff6b42b48bea1dd698eaa6a4741","target":"record","created_at":"2026-05-17T23:59:01Z","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":"8de42f7913e0b62786cc27f7574cb21c639fc53eb1a4995199bfb48f4ca9e21e","cross_cats_sorted":["cs.IT","cs.LG","math.IT","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-02-28T18:32:07Z","title_canon_sha256":"1c9308f141cfa8f54df5aeaf59a13c983bf78b0e6abbbf7a4107745aba10e3ba"},"schema_version":"1.0","source":{"id":"1802.10575","kind":"arxiv","version":2}},"canonical_sha256":"29f38b795562c5cb24a50d2accd4647efcc5bc187b1acd079032098333426553","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29f38b795562c5cb24a50d2accd4647efcc5bc187b1acd079032098333426553","first_computed_at":"2026-05-17T23:59:01.901678Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:01.901678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LS0Ha1TFNfUWy/jZ8aYSkUNeZCqSD2VPE+gKQvecFElD1UascYPaJ77LSs0necXNYu3ZMXOSk9DIKEyu4Py/CQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:01.902274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.10575","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49c824fb8ab21d995ab29bac948005dc0bd1cff6b42b48bea1dd698eaa6a4741","sha256:5db31d8ab49e2ac41eb50777381eb06fa144f6b07d274ceebb890712dc938632"],"state_sha256":"a76c736b5345134cfd1d0e7598367c27f23d5d81df40866686ef7bd6844c4fdf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DqH9JddH3y3WOqmxBBrJagiWVgW4UyN5dLL9cHBTC7riRNW1/3MK7vxJtDv2uRmAmHaIql2USSv/oe64nRWRAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T13:18:15.284755Z","bundle_sha256":"d78b469e0586c2ba9dcb8e25006696752bcc68013ef019ac5a588e2c8e66abe6"}}