{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZGGSHTR5CE4CAOXI4334EUPGAJ","short_pith_number":"pith:ZGGSHTR5","canonical_record":{"source":{"id":"1611.03473","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-11-10T20:32:48Z","cross_cats_sorted":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"title_canon_sha256":"b7b23663f7c43ae90da4967b9cc9231c9cd17602aa42002573104105dc05f9dd","abstract_canon_sha256":"42239afd81874d6b615598cd734e69fba040267bfde4af33e0e21ea2b8ee6541"},"schema_version":"1.0"},"canonical_sha256":"c98d23ce3d1138203ae8e6f7c251e602621503a2ce4f6d1b27f86c1ba6e2c793","source":{"kind":"arxiv","id":"1611.03473","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03473","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03473v2","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03473","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZGGSHTR5CE4C","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZGGSHTR5CE4CAOXI","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZGGSHTR5","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZGGSHTR5CE4CAOXI4334EUPGAJ","target":"record","payload":{"canonical_record":{"source":{"id":"1611.03473","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-11-10T20:32:48Z","cross_cats_sorted":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"title_canon_sha256":"b7b23663f7c43ae90da4967b9cc9231c9cd17602aa42002573104105dc05f9dd","abstract_canon_sha256":"42239afd81874d6b615598cd734e69fba040267bfde4af33e0e21ea2b8ee6541"},"schema_version":"1.0"},"canonical_sha256":"c98d23ce3d1138203ae8e6f7c251e602621503a2ce4f6d1b27f86c1ba6e2c793","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:18.738424Z","signature_b64":"JHUqXv2bUS6YcnIfpCxtBQdLo0cBvBixL8k53NzN3yFDmgmjCTA4ayi5pSjKFPwoKfC/qSFk+mu+tfZhUGqeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c98d23ce3d1138203ae8e6f7c251e602621503a2ce4f6d1b27f86c1ba6e2c793","last_reissued_at":"2026-05-18T00:44:18.737838Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:18.737838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.03473","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":"2026-05-18T00:44:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ho5yuBtE5XvObUt7h8WSqkTDdQykJNlGKcvuVSjN3d0zxfLls4Dqs14Ix9H9rsz+Wtfr4/w1YajMbDvfYL5Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T16:49:35.707444Z"},"content_sha256":"02d3565e86bf5cc0f9ff3d758cfacc657a0e70a684a40ac54e40a81b86038553","schema_version":"1.0","event_id":"sha256:02d3565e86bf5cc0f9ff3d758cfacc657a0e70a684a40ac54e40a81b86038553"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZGGSHTR5CE4CAOXI4334EUPGAJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"primary_cat":"cs.LG","authors_text":"Alistair Stewart, Daniel M. Kane, Ilias Diakonikolas","submitted_at":"2016-11-10T20:32:48Z","abstract_excerpt":"We describe a general technique that yields the first {\\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a {\\em super-polynomial gap} between the (information-theoretic) sample complexity and the computational complexity of {\\em any} Statistical Query algorithm for the problem. Our SQ lower bound for Problem (1) is q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03473","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":"2026-05-18T00:44:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qFBntH9CUJPiNKVgWHl3zZXa//0JEJfouPsjnshVp3XpGZM8rbvo8UvjRtjQoD9tOlN5VaVj4jsHwYKcG9R0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T16:49:35.707803Z"},"content_sha256":"b23fcae54211237e1819c0341da3756a875c1268a186ee673776b4572093c84d","schema_version":"1.0","event_id":"sha256:b23fcae54211237e1819c0341da3756a875c1268a186ee673776b4572093c84d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/bundle.json","state_url":"https://pith.science/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/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":"2026-06-05T16:49:35Z","links":{"resolver":"https://pith.science/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ","bundle":"https://pith.science/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/bundle.json","state":"https://pith.science/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGGSHTR5CE4CAOXI4334EUPGAJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZGGSHTR5CE4CAOXI4334EUPGAJ","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":"42239afd81874d6b615598cd734e69fba040267bfde4af33e0e21ea2b8ee6541","cross_cats_sorted":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-11-10T20:32:48Z","title_canon_sha256":"b7b23663f7c43ae90da4967b9cc9231c9cd17602aa42002573104105dc05f9dd"},"schema_version":"1.0","source":{"id":"1611.03473","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03473","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03473v2","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03473","created_at":"2026-05-18T00:44:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZGGSHTR5CE4C","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZGGSHTR5CE4CAOXI","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZGGSHTR5","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:b23fcae54211237e1819c0341da3756a875c1268a186ee673776b4572093c84d","target":"graph","created_at":"2026-05-18T00:44:18Z","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 describe a general technique that yields the first {\\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a {\\em super-polynomial gap} between the (information-theoretic) sample complexity and the computational complexity of {\\em any} Statistical Query algorithm for the problem. Our SQ lower bound for Problem (1) is q","authors_text":"Alistair Stewart, Daniel M. Kane, Ilias Diakonikolas","cross_cats":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-11-10T20:32:48Z","title":"Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03473","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":"sha256:02d3565e86bf5cc0f9ff3d758cfacc657a0e70a684a40ac54e40a81b86038553","target":"record","created_at":"2026-05-18T00:44:18Z","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":"42239afd81874d6b615598cd734e69fba040267bfde4af33e0e21ea2b8ee6541","cross_cats_sorted":["cs.CC","cs.DS","cs.IT","math.IT","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2016-11-10T20:32:48Z","title_canon_sha256":"b7b23663f7c43ae90da4967b9cc9231c9cd17602aa42002573104105dc05f9dd"},"schema_version":"1.0","source":{"id":"1611.03473","kind":"arxiv","version":2}},"canonical_sha256":"c98d23ce3d1138203ae8e6f7c251e602621503a2ce4f6d1b27f86c1ba6e2c793","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c98d23ce3d1138203ae8e6f7c251e602621503a2ce4f6d1b27f86c1ba6e2c793","first_computed_at":"2026-05-18T00:44:18.737838Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:18.737838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JHUqXv2bUS6YcnIfpCxtBQdLo0cBvBixL8k53NzN3yFDmgmjCTA4ayi5pSjKFPwoKfC/qSFk+mu+tfZhUGqeAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:18.738424Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.03473","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02d3565e86bf5cc0f9ff3d758cfacc657a0e70a684a40ac54e40a81b86038553","sha256:b23fcae54211237e1819c0341da3756a875c1268a186ee673776b4572093c84d"],"state_sha256":"bf1d06ed3758ff3287001f90e0554b6de891206646deecb448f7a119e9fb8572"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZLdTIK5/4F4NPPBFcvsbFth24F0J426oQBScJBB6ouCGDbMDOUjVIYQqp8ZCZbjuYNwh3qlAg6XiH0qzTA5cBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T16:49:35.709752Z","bundle_sha256":"c352fc34905b5ef4961672fb251dba76a0688955ea9e0cc18c90b9fdf7c4d4b2"}}