{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:JDDAW7NN6M4RJZVVQ65RWDZIKC","short_pith_number":"pith:JDDAW7NN","canonical_record":{"source":{"id":"1311.4025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-11-16T06:53:44Z","cross_cats_sorted":[],"title_canon_sha256":"6b2118f78c43bc605e96027214c38afc2ffbfb8ff2e032aa7c3739df23639931","abstract_canon_sha256":"5ad1a13eb4018c4236c24af9e72f7b7d0b35427318f6316139df26a533bae504"},"schema_version":"1.0"},"canonical_sha256":"48c60b7dadf33914e6b587bb1b0f2850a2a7ffadf3b4e011a241fe248443054a","source":{"kind":"arxiv","id":"1311.4025","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4025","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4025v3","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4025","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"pith_short_12","alias_value":"JDDAW7NN6M4R","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JDDAW7NN6M4RJZVV","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JDDAW7NN","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:JDDAW7NN6M4RJZVVQ65RWDZIKC","target":"record","payload":{"canonical_record":{"source":{"id":"1311.4025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-11-16T06:53:44Z","cross_cats_sorted":[],"title_canon_sha256":"6b2118f78c43bc605e96027214c38afc2ffbfb8ff2e032aa7c3739df23639931","abstract_canon_sha256":"5ad1a13eb4018c4236c24af9e72f7b7d0b35427318f6316139df26a533bae504"},"schema_version":"1.0"},"canonical_sha256":"48c60b7dadf33914e6b587bb1b0f2850a2a7ffadf3b4e011a241fe248443054a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:35.428949Z","signature_b64":"131VuczjXcjUZL6uAslQCtftONSAIYAKtzEdzt9lNmiQnZInOt31CmD4PBgceYEny3FDwkd1WzHSXQfhCzMBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48c60b7dadf33914e6b587bb1b0f2850a2a7ffadf3b4e011a241fe248443054a","last_reissued_at":"2026-05-18T02:57:35.428239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:35.428239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.4025","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jIPEN+TR4ubu3OsKayd9YqyxTE/lThOEq1Ob0RLAByX5lI7H+VG/kqdHbR194M0SrCxJJbpsLndSy6D+Pa0jDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T00:01:15.629742Z"},"content_sha256":"be99b869a94639df78238a83ba94721001a3ba54bc222554533cef6fe8e06989","schema_version":"1.0","event_id":"sha256:be99b869a94639df78238a83ba94721001a3ba54bc222554533cef6fe8e06989"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:JDDAW7NN6M4RJZVVQ65RWDZIKC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Signal Recovery from Pooling Representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ML","authors_text":"Arthur Szlam, Joan Bruna, Yann LeCun","submitted_at":"2013-11-16T06:53:44Z","abstract_excerpt":"In this work we compute lower Lipschitz bounds of $\\ell_p$ pooling operators for $p=1, 2, \\infty$ as well as $\\ell_p$ pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm that pooling layers can be inverted with phase recovery algorithms. Moreover, the regularity of the inverse pooling, controlled by the lower Lipschitz constant, is empirically verified with a nearest neighbor regression."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4025","kind":"arxiv","version":3},"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":"2026-05-18T02:57:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NEYMv4HgXAkOCXRQW8Ydqew7jLJW8l99Z7f3dB75aYIU5B6VpjsTwXnLzJLqwd5ILJNNWSLHuxeom8E7VK+FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T00:01:15.630364Z"},"content_sha256":"735c014fe20e7fbed691e6152496bbbe9ac2f882746b42723afce72338f540e7","schema_version":"1.0","event_id":"sha256:735c014fe20e7fbed691e6152496bbbe9ac2f882746b42723afce72338f540e7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/bundle.json","state_url":"https://pith.science/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/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":"2026-08-15T00:01:15Z","links":{"resolver":"https://pith.science/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC","bundle":"https://pith.science/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/bundle.json","state":"https://pith.science/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JDDAW7NN6M4RJZVVQ65RWDZIKC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:JDDAW7NN6M4RJZVVQ65RWDZIKC","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":"5ad1a13eb4018c4236c24af9e72f7b7d0b35427318f6316139df26a533bae504","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-11-16T06:53:44Z","title_canon_sha256":"6b2118f78c43bc605e96027214c38afc2ffbfb8ff2e032aa7c3739df23639931"},"schema_version":"1.0","source":{"id":"1311.4025","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4025","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4025v3","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4025","created_at":"2026-05-18T02:57:35Z"},{"alias_kind":"pith_short_12","alias_value":"JDDAW7NN6M4R","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JDDAW7NN6M4RJZVV","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JDDAW7NN","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:735c014fe20e7fbed691e6152496bbbe9ac2f882746b42723afce72338f540e7","target":"graph","created_at":"2026-05-18T02:57:35Z","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":"In this work we compute lower Lipschitz bounds of $\\ell_p$ pooling operators for $p=1, 2, \\infty$ as well as $\\ell_p$ pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm that pooling layers can be inverted with phase recovery algorithms. Moreover, the regularity of the inverse pooling, controlled by the lower Lipschitz constant, is empirically verified with a nearest neighbor regression.","authors_text":"Arthur Szlam, Joan Bruna, Yann LeCun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-11-16T06:53:44Z","title":"Signal Recovery from Pooling Representations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4025","kind":"arxiv","version":3},"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:be99b869a94639df78238a83ba94721001a3ba54bc222554533cef6fe8e06989","target":"record","created_at":"2026-05-18T02:57:35Z","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":"5ad1a13eb4018c4236c24af9e72f7b7d0b35427318f6316139df26a533bae504","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2013-11-16T06:53:44Z","title_canon_sha256":"6b2118f78c43bc605e96027214c38afc2ffbfb8ff2e032aa7c3739df23639931"},"schema_version":"1.0","source":{"id":"1311.4025","kind":"arxiv","version":3}},"canonical_sha256":"48c60b7dadf33914e6b587bb1b0f2850a2a7ffadf3b4e011a241fe248443054a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48c60b7dadf33914e6b587bb1b0f2850a2a7ffadf3b4e011a241fe248443054a","first_computed_at":"2026-05-18T02:57:35.428239Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:35.428239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"131VuczjXcjUZL6uAslQCtftONSAIYAKtzEdzt9lNmiQnZInOt31CmD4PBgceYEny3FDwkd1WzHSXQfhCzMBCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:35.428949Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.4025","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be99b869a94639df78238a83ba94721001a3ba54bc222554533cef6fe8e06989","sha256:735c014fe20e7fbed691e6152496bbbe9ac2f882746b42723afce72338f540e7"],"state_sha256":"6ac1cfa692c89e34f9d91c724096ce3a393cda3f1b42782c96c069c1e5fa20c2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s+u0lPSMNTrX+LBc9Dmb8g84W7Y3mVjp4SO1N52yeagwPpSrZI0M6Nh4MJlE4CKODixiiMHPHxciR0sRoEc6AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T00:01:15.639545Z","bundle_sha256":"05b147da486d7b33e54119317fadf1833601e65a94d03b136eeb34f7e964a14d"}}