{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5CDGHHVYDE5L5M4NHQTE4VMYUG","short_pith_number":"pith:5CDGHHVY","canonical_record":{"source":{"id":"1601.00948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-05T20:00:08Z","cross_cats_sorted":["math.OA","math.SP"],"title_canon_sha256":"ef5a8d7203aefe97481ba88500fef5375538f720c9e3dfc62059ad9ac251a78e","abstract_canon_sha256":"5dde3c310c2755be4f280188678a6acf6950a3d77fea2a398c1fd8b610fb3c5c"},"schema_version":"1.0"},"canonical_sha256":"e886639eb8193abeb38d3c264e5598a1a7269a4a2159f0da3992fb7557032214","source":{"kind":"arxiv","id":"1601.00948","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00948","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00948v2","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00948","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"pith_short_12","alias_value":"5CDGHHVYDE5L","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5CDGHHVYDE5L5M4N","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5CDGHHVY","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5CDGHHVYDE5L5M4NHQTE4VMYUG","target":"record","payload":{"canonical_record":{"source":{"id":"1601.00948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-05T20:00:08Z","cross_cats_sorted":["math.OA","math.SP"],"title_canon_sha256":"ef5a8d7203aefe97481ba88500fef5375538f720c9e3dfc62059ad9ac251a78e","abstract_canon_sha256":"5dde3c310c2755be4f280188678a6acf6950a3d77fea2a398c1fd8b610fb3c5c"},"schema_version":"1.0"},"canonical_sha256":"e886639eb8193abeb38d3c264e5598a1a7269a4a2159f0da3992fb7557032214","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:38.048122Z","signature_b64":"naCu238pDiHPNYmlA7cb4CJdox7QWNDXDR/NPb7vNgNM+2O3zs0o84Kkt6of5bYhfKFB7wgHQkbzQXsBVjBkBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e886639eb8193abeb38d3c264e5598a1a7269a4a2159f0da3992fb7557032214","last_reissued_at":"2026-05-18T00:56:38.047384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:38.047384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.00948","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":"2026-05-18T00:56:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6+8P3kjH5eNw5lHYk8fp1UwDAqHyh6/rTf/74/buiQSatojnhkAVp+1C3upgABBVhOd/AJmctYwoz4dnU8GcDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T17:05:12.140946Z"},"content_sha256":"d62eba880b74a64e0d308f00932ab2a77b6d6647e3909de1858bc7e71f10c2eb","schema_version":"1.0","event_id":"sha256:d62eba880b74a64e0d308f00932ab2a77b6d6647e3909de1858bc7e71f10c2eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5CDGHHVYDE5L5M4NHQTE4VMYUG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Restricted invertibility revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA","math.SP"],"primary_cat":"math.FA","authors_text":"Assaf Naor, Pierre Youssef","submitted_at":"2016-01-05T20:00:08Z","abstract_excerpt":"Suppose that $m,n\\in \\mathbb{N}$ and that $A:\\mathbb{R}^m\\to \\mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\\in \\mathbb{N}$ satisfy $k<r\\le \\mathrm{\\bf rank(A)}$ then there exists a subset $\\sigma\\subseteq \\{1,\\ldots,m\\}$ with $|\\sigma|=k$ such that the restriction of $A$ to $\\mathbb{R}^{\\sigma}\\subseteq \\mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\\mathbb{R}^{\\sigma})\\to \\mathbb{R}^m$ is at most a constant multiple of the quantity $\\sqrt{mr/((r-k)\\sum_{i=r}^m \\mathsf{s}_i(A)^2)}$, where $\\mathsf{s}_1(A)\\geqslant\\ldots\\geqslant \\mathsf{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00948","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":"2026-05-18T00:56:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NlN2TARAlju0Te1TAjGvRW53/eEk/Fnv+I5km3d0lBRrKxeU4WuW7QG4UyiaNVfRFTU6Zd79qi4ZACozbP55BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T17:05:12.141289Z"},"content_sha256":"2570c37efcd7aa786d359efe18f028994e058737336ad828f89f72771fbd9b95","schema_version":"1.0","event_id":"sha256:2570c37efcd7aa786d359efe18f028994e058737336ad828f89f72771fbd9b95"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/bundle.json","state_url":"https://pith.science/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/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":"2026-06-20T17:05:12Z","links":{"resolver":"https://pith.science/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG","bundle":"https://pith.science/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/bundle.json","state":"https://pith.science/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5CDGHHVYDE5L5M4NHQTE4VMYUG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5CDGHHVYDE5L5M4NHQTE4VMYUG","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":"5dde3c310c2755be4f280188678a6acf6950a3d77fea2a398c1fd8b610fb3c5c","cross_cats_sorted":["math.OA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-05T20:00:08Z","title_canon_sha256":"ef5a8d7203aefe97481ba88500fef5375538f720c9e3dfc62059ad9ac251a78e"},"schema_version":"1.0","source":{"id":"1601.00948","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00948","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00948v2","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00948","created_at":"2026-05-18T00:56:38Z"},{"alias_kind":"pith_short_12","alias_value":"5CDGHHVYDE5L","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5CDGHHVYDE5L5M4N","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5CDGHHVY","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:2570c37efcd7aa786d359efe18f028994e058737336ad828f89f72771fbd9b95","target":"graph","created_at":"2026-05-18T00:56:38Z","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":"Suppose that $m,n\\in \\mathbb{N}$ and that $A:\\mathbb{R}^m\\to \\mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\\in \\mathbb{N}$ satisfy $k<r\\le \\mathrm{\\bf rank(A)}$ then there exists a subset $\\sigma\\subseteq \\{1,\\ldots,m\\}$ with $|\\sigma|=k$ such that the restriction of $A$ to $\\mathbb{R}^{\\sigma}\\subseteq \\mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\\mathbb{R}^{\\sigma})\\to \\mathbb{R}^m$ is at most a constant multiple of the quantity $\\sqrt{mr/((r-k)\\sum_{i=r}^m \\mathsf{s}_i(A)^2)}$, where $\\mathsf{s}_1(A)\\geqslant\\ldots\\geqslant \\mathsf{","authors_text":"Assaf Naor, Pierre Youssef","cross_cats":["math.OA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-05T20:00:08Z","title":"Restricted invertibility revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00948","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:d62eba880b74a64e0d308f00932ab2a77b6d6647e3909de1858bc7e71f10c2eb","target":"record","created_at":"2026-05-18T00:56:38Z","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":"5dde3c310c2755be4f280188678a6acf6950a3d77fea2a398c1fd8b610fb3c5c","cross_cats_sorted":["math.OA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-01-05T20:00:08Z","title_canon_sha256":"ef5a8d7203aefe97481ba88500fef5375538f720c9e3dfc62059ad9ac251a78e"},"schema_version":"1.0","source":{"id":"1601.00948","kind":"arxiv","version":2}},"canonical_sha256":"e886639eb8193abeb38d3c264e5598a1a7269a4a2159f0da3992fb7557032214","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e886639eb8193abeb38d3c264e5598a1a7269a4a2159f0da3992fb7557032214","first_computed_at":"2026-05-18T00:56:38.047384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:38.047384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"naCu238pDiHPNYmlA7cb4CJdox7QWNDXDR/NPb7vNgNM+2O3zs0o84Kkt6of5bYhfKFB7wgHQkbzQXsBVjBkBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:38.048122Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.00948","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d62eba880b74a64e0d308f00932ab2a77b6d6647e3909de1858bc7e71f10c2eb","sha256:2570c37efcd7aa786d359efe18f028994e058737336ad828f89f72771fbd9b95"],"state_sha256":"001815a6314dee19d518a715220889db560bb7118c19f0439f4b5f34c3ab18ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"108+cEMBgMz215fkM+EanQXx7ITBk19UoIjL8zG78TnWuJWVLK4U4bRpPQFVZVtEwRuHAH1ekbRuRCL4xUOJAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T17:05:12.143264Z","bundle_sha256":"64abb8c3581771c81a494635fa02410100404a22c56da70e73fe964cd0e8d24c"}}