{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:R4LJGHPXMTK57IVMD2FO2QXETP","short_pith_number":"pith:R4LJGHPX","canonical_record":{"source":{"id":"0912.1403","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2009-12-08T04:54:17Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"7913dbcde2aee687c5842c57d5e1d6e0f0791bc7b1550a0fbd5eca3acfc54ebc","abstract_canon_sha256":"36093c427a9edfcdf98bd24b883d52b5456d47dc819cb70e6c3dfc3a02067459"},"schema_version":"1.0"},"canonical_sha256":"8f16931df764d5dfa2ac1e8aed42e49bd37594dac1c8c3ced9baa876f57e9a07","source":{"kind":"arxiv","id":"0912.1403","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1403","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1403v2","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1403","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"pith_short_12","alias_value":"R4LJGHPXMTK5","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"R4LJGHPXMTK57IVM","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"R4LJGHPX","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:R4LJGHPXMTK57IVMD2FO2QXETP","target":"record","payload":{"canonical_record":{"source":{"id":"0912.1403","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2009-12-08T04:54:17Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"7913dbcde2aee687c5842c57d5e1d6e0f0791bc7b1550a0fbd5eca3acfc54ebc","abstract_canon_sha256":"36093c427a9edfcdf98bd24b883d52b5456d47dc819cb70e6c3dfc3a02067459"},"schema_version":"1.0"},"canonical_sha256":"8f16931df764d5dfa2ac1e8aed42e49bd37594dac1c8c3ced9baa876f57e9a07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:19.598846Z","signature_b64":"FEiKdJpv6wCFQ+lWdif5iDFknEzp+yAQf/S5R5HQJ1nrQa++VxBNYLuj/h3Q7eV9s8a0A/zznbK4qpOw/t9vCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f16931df764d5dfa2ac1e8aed42e49bd37594dac1c8c3ced9baa876f57e9a07","last_reissued_at":"2026-05-18T04:32:19.598158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:19.598158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.1403","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":"2026-05-18T04:32:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+fDbjudcZYh62L5ovIEn6TSVnpUrzjIOfEMGM1napU6erpm3e2RcSlyGAdiieEXNLdJJIqKNMYcgCOCbg1A9Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T12:56:46.744013Z"},"content_sha256":"e40c1d5d8e8c5a85dfdf6b31a515bc461e39d05b40d7bb0d22340584a1616be1","schema_version":"1.0","event_id":"sha256:e40c1d5d8e8c5a85dfdf6b31a515bc461e39d05b40d7bb0d22340584a1616be1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:R4LJGHPXMTK57IVMD2FO2QXETP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algorithms and Hardness for Subspace Approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Amit Deshpande, Kasturi Varadarajan, Madhur Tulsiani, Nisheeth K. Vishnoi","submitted_at":"2009-12-08T04:54:17Z","abstract_excerpt":"The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\\ell_{p}$ norm instead. Most of the previous work on subspace approximation has focused on small or constant $k$ and $p$, using coresets and sampling techniques from computational geometry.\n  In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, \\emph{for any $k$ and any $p \\geq 2$}, with the approximat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1403","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":"2026-05-18T04:32:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HpBbl7ViExJ+LMsYbew5Erc0wN5RScKwcc2pi4TDlDs0XXbo47XMAKEqFR7/L2Ls5iD0SSdx6QTi71VxYTArBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T12:56:46.744392Z"},"content_sha256":"60fa2de3e941a580ea24bd252b8a601eec8e884eb4b7e8a2ce1f76c341eb5afb","schema_version":"1.0","event_id":"sha256:60fa2de3e941a580ea24bd252b8a601eec8e884eb4b7e8a2ce1f76c341eb5afb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R4LJGHPXMTK57IVMD2FO2QXETP/bundle.json","state_url":"https://pith.science/pith/R4LJGHPXMTK57IVMD2FO2QXETP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R4LJGHPXMTK57IVMD2FO2QXETP/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":"2026-06-08T12:56:46Z","links":{"resolver":"https://pith.science/pith/R4LJGHPXMTK57IVMD2FO2QXETP","bundle":"https://pith.science/pith/R4LJGHPXMTK57IVMD2FO2QXETP/bundle.json","state":"https://pith.science/pith/R4LJGHPXMTK57IVMD2FO2QXETP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R4LJGHPXMTK57IVMD2FO2QXETP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:R4LJGHPXMTK57IVMD2FO2QXETP","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":"36093c427a9edfcdf98bd24b883d52b5456d47dc819cb70e6c3dfc3a02067459","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2009-12-08T04:54:17Z","title_canon_sha256":"7913dbcde2aee687c5842c57d5e1d6e0f0791bc7b1550a0fbd5eca3acfc54ebc"},"schema_version":"1.0","source":{"id":"0912.1403","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1403","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1403v2","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1403","created_at":"2026-05-18T04:32:19Z"},{"alias_kind":"pith_short_12","alias_value":"R4LJGHPXMTK5","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"R4LJGHPXMTK57IVM","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"R4LJGHPX","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:60fa2de3e941a580ea24bd252b8a601eec8e884eb4b7e8a2ce1f76c341eb5afb","target":"graph","created_at":"2026-05-18T04:32:19Z","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":"The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\\ell_{p}$ norm instead. Most of the previous work on subspace approximation has focused on small or constant $k$ and $p$, using coresets and sampling techniques from computational geometry.\n  In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, \\emph{for any $k$ and any $p \\geq 2$}, with the approximat","authors_text":"Amit Deshpande, Kasturi Varadarajan, Madhur Tulsiani, Nisheeth K. Vishnoi","cross_cats":["cs.CC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2009-12-08T04:54:17Z","title":"Algorithms and Hardness for Subspace Approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1403","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:e40c1d5d8e8c5a85dfdf6b31a515bc461e39d05b40d7bb0d22340584a1616be1","target":"record","created_at":"2026-05-18T04:32:19Z","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":"36093c427a9edfcdf98bd24b883d52b5456d47dc819cb70e6c3dfc3a02067459","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2009-12-08T04:54:17Z","title_canon_sha256":"7913dbcde2aee687c5842c57d5e1d6e0f0791bc7b1550a0fbd5eca3acfc54ebc"},"schema_version":"1.0","source":{"id":"0912.1403","kind":"arxiv","version":2}},"canonical_sha256":"8f16931df764d5dfa2ac1e8aed42e49bd37594dac1c8c3ced9baa876f57e9a07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f16931df764d5dfa2ac1e8aed42e49bd37594dac1c8c3ced9baa876f57e9a07","first_computed_at":"2026-05-18T04:32:19.598158Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:32:19.598158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FEiKdJpv6wCFQ+lWdif5iDFknEzp+yAQf/S5R5HQJ1nrQa++VxBNYLuj/h3Q7eV9s8a0A/zznbK4qpOw/t9vCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:32:19.598846Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.1403","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e40c1d5d8e8c5a85dfdf6b31a515bc461e39d05b40d7bb0d22340584a1616be1","sha256:60fa2de3e941a580ea24bd252b8a601eec8e884eb4b7e8a2ce1f76c341eb5afb"],"state_sha256":"54667ddc2445944d2a21eced18d6a74b53481463de7788706077b729c9650b12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WiwvEQbb1ASzUezlBRNK2cTolo+wh88nMsO+1J/TbB8na6lX5fzYYqzKmtVLHW5OY6lA/Ee9D/AIBd7mr/rnCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T12:56:46.746300Z","bundle_sha256":"b8ac98ef4c98f75f7cf977bee478fbf3d51c253494cf8d6774cf476d88b0ced6"}}