{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:TDL62HZP3VVJXPKWPC4B2XQXLH","short_pith_number":"pith:TDL62HZP","canonical_record":{"source":{"id":"2412.19677","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-12-27T14:57:40Z","cross_cats_sorted":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"title_canon_sha256":"1e3b4e300d1f73d0f31ae60cf7126d80e9ad6d49e001c2929b4f2da430a6a70c","abstract_canon_sha256":"3584ee11cb1f35f0e953e02d981c9934130acf6322118d844735cb232823ac8e"},"schema_version":"1.0"},"canonical_sha256":"98d7ed1f2fdd6a9bbd5678b81d5e1759e35fa41ed7c5cb07e4dac0befc688da4","source":{"kind":"arxiv","id":"2412.19677","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.19677","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"2412.19677v1","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.19677","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"TDL62HZP3VVJ","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_16","alias_value":"TDL62HZP3VVJXPKW","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_8","alias_value":"TDL62HZP","created_at":"2026-07-05T09:54:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:TDL62HZP3VVJXPKWPC4B2XQXLH","target":"record","payload":{"canonical_record":{"source":{"id":"2412.19677","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-12-27T14:57:40Z","cross_cats_sorted":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"title_canon_sha256":"1e3b4e300d1f73d0f31ae60cf7126d80e9ad6d49e001c2929b4f2da430a6a70c","abstract_canon_sha256":"3584ee11cb1f35f0e953e02d981c9934130acf6322118d844735cb232823ac8e"},"schema_version":"1.0"},"canonical_sha256":"98d7ed1f2fdd6a9bbd5678b81d5e1759e35fa41ed7c5cb07e4dac0befc688da4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:54:39.555493Z","signature_b64":"JnqyucN4LUFconOdGdW4GJ069dXUHGYdHw94gl6jJfLubr3TIH83inM9hVGXE+t7Z/ZnxfMvINiZ4Eu8MdAGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98d7ed1f2fdd6a9bbd5678b81d5e1759e35fa41ed7c5cb07e4dac0befc688da4","last_reissued_at":"2026-07-05T09:54:39.554982Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:54:39.554982Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.19677","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":"2026-07-05T09:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t/q1jxP8q9CVsbZ3Ghk789iKpgoEvAY1/qM8DOy6bYs1BFUTGsoIlg+a7Aevale4UsmX8MmDV0WXn5Dc3YdxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:39:45.199652Z"},"content_sha256":"2a3955a138000e725a1e1fe8636b7163ae30a1a4014567ca814ef08a0f9c5080","schema_version":"1.0","event_id":"sha256:2a3955a138000e725a1e1fe8636b7163ae30a1a4014567ca814ef08a0f9c5080"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:TDL62HZP3VVJXPKWPC4B2XQXLH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deep ReLU networks -- injectivity capacity upper bounds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"primary_cat":"stat.ML","authors_text":"Mihailo Stojnic","submitted_at":"2024-12-27T14:57:40Z","abstract_excerpt":"We study deep ReLU feed forward neural networks (NN) and their injectivity abilities. The main focus is on \\emph{precisely} determining the so-called injectivity capacity. For any given hidden layers architecture, it is defined as the minimal ratio between number of network's outputs and inputs which ensures unique recoverability of the input from a realizable output. A strong recent progress in precisely studying single ReLU layer injectivity properties is here moved to a deep network level. In particular, we develop a program that connects deep $l$-layer net injectivity to an $l$-extension o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19677","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/2412.19677/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":"2026-07-05T09:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CEs1qKG1JvxoWg8hEd4q7qBX+pp+79CIdB3Ca3cBPH2Fot5/uNG8to/m4Vs+axiAwJN53OnPSh/lRrneEH1QCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:39:45.200423Z"},"content_sha256":"e348d680504257e8dffe6e4d04d7070677be3c1a3fe551f3cbf3003130ff0beb","schema_version":"1.0","event_id":"sha256:e348d680504257e8dffe6e4d04d7070677be3c1a3fe551f3cbf3003130ff0beb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/bundle.json","state_url":"https://pith.science/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/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":"2026-08-12T08:39:45Z","links":{"resolver":"https://pith.science/pith/TDL62HZP3VVJXPKWPC4B2XQXLH","bundle":"https://pith.science/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/bundle.json","state":"https://pith.science/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TDL62HZP3VVJXPKWPC4B2XQXLH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:TDL62HZP3VVJXPKWPC4B2XQXLH","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":"3584ee11cb1f35f0e953e02d981c9934130acf6322118d844735cb232823ac8e","cross_cats_sorted":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-12-27T14:57:40Z","title_canon_sha256":"1e3b4e300d1f73d0f31ae60cf7126d80e9ad6d49e001c2929b4f2da430a6a70c"},"schema_version":"1.0","source":{"id":"2412.19677","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.19677","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"2412.19677v1","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.19677","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"TDL62HZP3VVJ","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_16","alias_value":"TDL62HZP3VVJXPKW","created_at":"2026-07-05T09:54:39Z"},{"alias_kind":"pith_short_8","alias_value":"TDL62HZP","created_at":"2026-07-05T09:54:39Z"}],"graph_snapshots":[{"event_id":"sha256:e348d680504257e8dffe6e4d04d7070677be3c1a3fe551f3cbf3003130ff0beb","target":"graph","created_at":"2026-07-05T09:54:39Z","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/2412.19677/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study deep ReLU feed forward neural networks (NN) and their injectivity abilities. The main focus is on \\emph{precisely} determining the so-called injectivity capacity. For any given hidden layers architecture, it is defined as the minimal ratio between number of network's outputs and inputs which ensures unique recoverability of the input from a realizable output. A strong recent progress in precisely studying single ReLU layer injectivity properties is here moved to a deep network level. In particular, we develop a program that connects deep $l$-layer net injectivity to an $l$-extension o","authors_text":"Mihailo Stojnic","cross_cats":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-12-27T14:57:40Z","title":"Deep ReLU networks -- injectivity capacity upper bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19677","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":"sha256:2a3955a138000e725a1e1fe8636b7163ae30a1a4014567ca814ef08a0f9c5080","target":"record","created_at":"2026-07-05T09:54:39Z","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":"3584ee11cb1f35f0e953e02d981c9934130acf6322118d844735cb232823ac8e","cross_cats_sorted":["cond-mat.dis-nn","cs.IT","cs.LG","math.IT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-12-27T14:57:40Z","title_canon_sha256":"1e3b4e300d1f73d0f31ae60cf7126d80e9ad6d49e001c2929b4f2da430a6a70c"},"schema_version":"1.0","source":{"id":"2412.19677","kind":"arxiv","version":1}},"canonical_sha256":"98d7ed1f2fdd6a9bbd5678b81d5e1759e35fa41ed7c5cb07e4dac0befc688da4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98d7ed1f2fdd6a9bbd5678b81d5e1759e35fa41ed7c5cb07e4dac0befc688da4","first_computed_at":"2026-07-05T09:54:39.554982Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:54:39.554982Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JnqyucN4LUFconOdGdW4GJ069dXUHGYdHw94gl6jJfLubr3TIH83inM9hVGXE+t7Z/ZnxfMvINiZ4Eu8MdAGDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:54:39.555493Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.19677","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a3955a138000e725a1e1fe8636b7163ae30a1a4014567ca814ef08a0f9c5080","sha256:e348d680504257e8dffe6e4d04d7070677be3c1a3fe551f3cbf3003130ff0beb"],"state_sha256":"8ade76143246bde9ad82258e05bc8b34619271376a981fb2c423855a3b66ae52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YmP7J6Ex1Wp0eqBk23EOhcLDQoUk8VV9w1KYEPzeLkpHqXAH9xP25T52nH444PSmBAnQy/NOdL6+tMDbfVzoDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:39:45.205594Z","bundle_sha256":"37a148af8cd29bbadf5a89bd55d043d8688703e7f2dc7d54f89b68ea593cc305"}}