{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:OZ65VJUGAVMBA4WS2S4IVUMI3F","short_pith_number":"pith:OZ65VJUG","canonical_record":{"source":{"id":"2012.07774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-14T18:14:08Z","cross_cats_sorted":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"title_canon_sha256":"3a2c26823e086be04c389f6bc191abd8aaf91b640fc5039a31f7732a0eb124fb","abstract_canon_sha256":"e2eec43601a8be4ae133bccf856712911b1c9cf7a15458f238615bb9fb6c13ed"},"schema_version":"1.0"},"canonical_sha256":"767ddaa68605581072d2d4b88ad188d95d11267dd355f20193324bebf78f431e","source":{"kind":"arxiv","id":"2012.07774","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2012.07774","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"arxiv_version","alias_value":"2012.07774v1","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.07774","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_12","alias_value":"OZ65VJUGAVMB","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_16","alias_value":"OZ65VJUGAVMBA4WS","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_8","alias_value":"OZ65VJUG","created_at":"2026-07-05T01:59:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:OZ65VJUGAVMBA4WS2S4IVUMI3F","target":"record","payload":{"canonical_record":{"source":{"id":"2012.07774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-14T18:14:08Z","cross_cats_sorted":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"title_canon_sha256":"3a2c26823e086be04c389f6bc191abd8aaf91b640fc5039a31f7732a0eb124fb","abstract_canon_sha256":"e2eec43601a8be4ae133bccf856712911b1c9cf7a15458f238615bb9fb6c13ed"},"schema_version":"1.0"},"canonical_sha256":"767ddaa68605581072d2d4b88ad188d95d11267dd355f20193324bebf78f431e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:59:24.031516Z","signature_b64":"+ReMv5vSBxsk6YACZKXcpPfixlYcJ452+bB+MoUATo6JxwEEX2flfvPFoRvwfh0u3elsfFyhs9yOpWyUOlkhCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"767ddaa68605581072d2d4b88ad188d95d11267dd355f20193324bebf78f431e","last_reissued_at":"2026-07-05T01:59:24.031105Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:59:24.031105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2012.07774","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":"2026-07-05T01:59:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h5a8gp77pX/p35bNDu3S9vdcgh1Kw/tfU+Z+zRdLyo3BKxi/4aV8GF9AGRC2urWSV8H9dszLj3+5ZSPFtnp1Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T00:09:10.556201Z"},"content_sha256":"497a8a6e4a49c9b37f22842b9cb09f6793ac72f1c08ab376893b50b299a163c3","schema_version":"1.0","event_id":"sha256:497a8a6e4a49c9b37f22842b9cb09f6793ac72f1c08ab376893b50b299a163c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:OZ65VJUGAVMBA4WS2S4IVUMI3F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"primary_cat":"cs.LG","authors_text":"Daniel M. Kane, Ilias Diakonikolas","submitted_at":"2020-12-14T18:14:08Z","abstract_excerpt":"Let $V$ be any vector space of multivariate degree-$d$ homogeneous polynomials with co-dimension at most $k$, and $S$ be the set of points where all polynomials in $V$ {\\em nearly} vanish. We establish a qualitatively optimal upper bound on the size of $\\epsilon$-covers for $S$, in the $\\ell_2$-norm. Roughly speaking, we show that there exists an $\\epsilon$-cover for $S$ of cardinality $M = (k/\\epsilon)^{O_d(k^{1/d})}$. Our result is constructive yielding an algorithm to compute such an $\\epsilon$-cover that runs in time $\\mathrm{poly}(M)$.\n  Building on our structural result, we obtain signif"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.07774","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2012.07774/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":"2026-07-05T01:59:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7Y7zV2BbyfXFW9OK9Epv08S0dxz1t5DW9HGPxC2HZaS//QgODmEMndnqhnXJNXMPsrN/FuHTJJNDXKgSxtYhDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T00:09:10.556803Z"},"content_sha256":"f67594209dff5fa1b85dbf33923d7f6ff0009a9e612b4817650ea500045c92aa","schema_version":"1.0","event_id":"sha256:f67594209dff5fa1b85dbf33923d7f6ff0009a9e612b4817650ea500045c92aa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/bundle.json","state_url":"https://pith.science/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/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":"2026-08-14T00:09:10Z","links":{"resolver":"https://pith.science/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F","bundle":"https://pith.science/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/bundle.json","state":"https://pith.science/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OZ65VJUGAVMBA4WS2S4IVUMI3F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:OZ65VJUGAVMBA4WS2S4IVUMI3F","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":"e2eec43601a8be4ae133bccf856712911b1c9cf7a15458f238615bb9fb6c13ed","cross_cats_sorted":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-14T18:14:08Z","title_canon_sha256":"3a2c26823e086be04c389f6bc191abd8aaf91b640fc5039a31f7732a0eb124fb"},"schema_version":"1.0","source":{"id":"2012.07774","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2012.07774","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"arxiv_version","alias_value":"2012.07774v1","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.07774","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_12","alias_value":"OZ65VJUGAVMB","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_16","alias_value":"OZ65VJUGAVMBA4WS","created_at":"2026-07-05T01:59:24Z"},{"alias_kind":"pith_short_8","alias_value":"OZ65VJUG","created_at":"2026-07-05T01:59:24Z"}],"graph_snapshots":[{"event_id":"sha256:f67594209dff5fa1b85dbf33923d7f6ff0009a9e612b4817650ea500045c92aa","target":"graph","created_at":"2026-07-05T01:59:24Z","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/2012.07774/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $V$ be any vector space of multivariate degree-$d$ homogeneous polynomials with co-dimension at most $k$, and $S$ be the set of points where all polynomials in $V$ {\\em nearly} vanish. We establish a qualitatively optimal upper bound on the size of $\\epsilon$-covers for $S$, in the $\\ell_2$-norm. Roughly speaking, we show that there exists an $\\epsilon$-cover for $S$ of cardinality $M = (k/\\epsilon)^{O_d(k^{1/d})}$. Our result is constructive yielding an algorithm to compute such an $\\epsilon$-cover that runs in time $\\mathrm{poly}(M)$.\n  Building on our structural result, we obtain signif","authors_text":"Daniel M. Kane, Ilias Diakonikolas","cross_cats":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-14T18:14:08Z","title":"Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.07774","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":"sha256:497a8a6e4a49c9b37f22842b9cb09f6793ac72f1c08ab376893b50b299a163c3","target":"record","created_at":"2026-07-05T01:59:24Z","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":"e2eec43601a8be4ae133bccf856712911b1c9cf7a15458f238615bb9fb6c13ed","cross_cats_sorted":["cs.CC","cs.DS","math.AG","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-14T18:14:08Z","title_canon_sha256":"3a2c26823e086be04c389f6bc191abd8aaf91b640fc5039a31f7732a0eb124fb"},"schema_version":"1.0","source":{"id":"2012.07774","kind":"arxiv","version":1}},"canonical_sha256":"767ddaa68605581072d2d4b88ad188d95d11267dd355f20193324bebf78f431e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"767ddaa68605581072d2d4b88ad188d95d11267dd355f20193324bebf78f431e","first_computed_at":"2026-07-05T01:59:24.031105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:59:24.031105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+ReMv5vSBxsk6YACZKXcpPfixlYcJ452+bB+MoUATo6JxwEEX2flfvPFoRvwfh0u3elsfFyhs9yOpWyUOlkhCA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:59:24.031516Z","signed_message":"canonical_sha256_bytes"},"source_id":"2012.07774","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:497a8a6e4a49c9b37f22842b9cb09f6793ac72f1c08ab376893b50b299a163c3","sha256:f67594209dff5fa1b85dbf33923d7f6ff0009a9e612b4817650ea500045c92aa"],"state_sha256":"2feea879863480c83a2e5379d33bf5bdece466b73c029e9752a039770b198189"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FDMnRhHxX9JqGuQTyQ8qO8rPDKDnumyj8RsSOBaA4hE5Axk9dGBq/WMjLgHAUSX7Tfe1ZYngiNnc4HnrgCStDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T00:09:10.594514Z","bundle_sha256":"5f1c203bb3025a9f3195d25981d1f64c54c9ea0ea0e54ce53be78659688e5a1e"}}