{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:NGARCQBGBUHASF2VYRWJVUM3VY","short_pith_number":"pith:NGARCQBG","canonical_record":{"source":{"id":"1709.06080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-16T18:59:17Z","cross_cats_sorted":["cs.AI","math.NA"],"title_canon_sha256":"781c4e1336d8dba232598ae9193ef560e0795c1d8b4b9ef09350227480d03a60","abstract_canon_sha256":"e306d61bcea30cc8b1e09d55ed58cbc8d7c16d7361b8cb2ca231c7349151bc81"},"schema_version":"1.0"},"canonical_sha256":"69811140260d0e091755c46c9ad19bae16a25e75fe0a6f66f3acb88bd4663b70","source":{"kind":"arxiv","id":"1709.06080","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06080","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06080v1","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06080","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"pith_short_12","alias_value":"NGARCQBGBUHA","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"NGARCQBGBUHASF2V","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"NGARCQBG","created_at":"2026-05-18T12:31:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:NGARCQBGBUHASF2VYRWJVUM3VY","target":"record","payload":{"canonical_record":{"source":{"id":"1709.06080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-16T18:59:17Z","cross_cats_sorted":["cs.AI","math.NA"],"title_canon_sha256":"781c4e1336d8dba232598ae9193ef560e0795c1d8b4b9ef09350227480d03a60","abstract_canon_sha256":"e306d61bcea30cc8b1e09d55ed58cbc8d7c16d7361b8cb2ca231c7349151bc81"},"schema_version":"1.0"},"canonical_sha256":"69811140260d0e091755c46c9ad19bae16a25e75fe0a6f66f3acb88bd4663b70","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:54.449612Z","signature_b64":"LBLo1gaJGFK29BVk9XprNuug1qtut4U2YsMiOGZGZTaCS/A5LoBopqhFzbooC7Z7tGuGOgehNf1V2iTHWRr+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69811140260d0e091755c46c9ad19bae16a25e75fe0a6f66f3acb88bd4663b70","last_reissued_at":"2026-05-18T00:34:54.448938Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:54.448938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.06080","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":"2026-05-18T00:34:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lYJ9TjYwZdnuIEhADcyL/XIzelmK0s3u/iC8O2UwuA/CKvD1NgWWc6R0MzOR8jqVrPM93lmNchZy0wPtkOgPDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:15:01.193371Z"},"content_sha256":"ef1dc74c409b3e11f9f623d31413295fdd2345bf98ab86213680ea86cd8f4f44","schema_version":"1.0","event_id":"sha256:ef1dc74c409b3e11f9f623d31413295fdd2345bf98ab86213680ea86cd8f4f44"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:NGARCQBGBUHASF2VYRWJVUM3VY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Feedforward and Recurrent Neural Networks Backward Propagation and Hessian in Matrix Form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","math.NA"],"primary_cat":"cs.LG","authors_text":"Maxim Naumov","submitted_at":"2017-09-16T18:59:17Z","abstract_excerpt":"In this paper we focus on the linear algebra theory behind feedforward (FNN) and recurrent (RNN) neural networks. We review backward propagation, including backward propagation through time (BPTT). Also, we obtain a new exact expression for Hessian, which represents second order effects. We show that for $t$ time steps the weight gradient can be expressed as a rank-$t$ matrix, while the weight Hessian is as a sum of $t^{2}$ Kronecker products of rank-$1$ and $W^{T}AW$ matrices, for some matrix $A$ and weight matrix $W$. Also, we show that for a mini-batch of size $r$, the weight update can be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06080","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":""},"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":"2026-05-18T00:34:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rOakoHvRw5LCXkmqprqbmyrT9IAuxpdpRLTKE6Z9vL0+pPG+YTSXaWVv918IFyh52mvtQiaLMw1ZM4wwHu4cDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:15:01.194064Z"},"content_sha256":"83da323f6393896c902cb772bfe7a11a5e2c5ea82901e7327ceb80fbb4ae9cf1","schema_version":"1.0","event_id":"sha256:83da323f6393896c902cb772bfe7a11a5e2c5ea82901e7327ceb80fbb4ae9cf1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NGARCQBGBUHASF2VYRWJVUM3VY/bundle.json","state_url":"https://pith.science/pith/NGARCQBGBUHASF2VYRWJVUM3VY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NGARCQBGBUHASF2VYRWJVUM3VY/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":"2026-05-26T13:15:01Z","links":{"resolver":"https://pith.science/pith/NGARCQBGBUHASF2VYRWJVUM3VY","bundle":"https://pith.science/pith/NGARCQBGBUHASF2VYRWJVUM3VY/bundle.json","state":"https://pith.science/pith/NGARCQBGBUHASF2VYRWJVUM3VY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NGARCQBGBUHASF2VYRWJVUM3VY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:NGARCQBGBUHASF2VYRWJVUM3VY","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":"e306d61bcea30cc8b1e09d55ed58cbc8d7c16d7361b8cb2ca231c7349151bc81","cross_cats_sorted":["cs.AI","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-16T18:59:17Z","title_canon_sha256":"781c4e1336d8dba232598ae9193ef560e0795c1d8b4b9ef09350227480d03a60"},"schema_version":"1.0","source":{"id":"1709.06080","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.06080","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"arxiv_version","alias_value":"1709.06080v1","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06080","created_at":"2026-05-18T00:34:54Z"},{"alias_kind":"pith_short_12","alias_value":"NGARCQBGBUHA","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"NGARCQBGBUHASF2V","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"NGARCQBG","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:83da323f6393896c902cb772bfe7a11a5e2c5ea82901e7327ceb80fbb4ae9cf1","target":"graph","created_at":"2026-05-18T00:34:54Z","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 paper we focus on the linear algebra theory behind feedforward (FNN) and recurrent (RNN) neural networks. We review backward propagation, including backward propagation through time (BPTT). Also, we obtain a new exact expression for Hessian, which represents second order effects. We show that for $t$ time steps the weight gradient can be expressed as a rank-$t$ matrix, while the weight Hessian is as a sum of $t^{2}$ Kronecker products of rank-$1$ and $W^{T}AW$ matrices, for some matrix $A$ and weight matrix $W$. Also, we show that for a mini-batch of size $r$, the weight update can be ","authors_text":"Maxim Naumov","cross_cats":["cs.AI","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-16T18:59:17Z","title":"Feedforward and Recurrent Neural Networks Backward Propagation and Hessian in Matrix Form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06080","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:ef1dc74c409b3e11f9f623d31413295fdd2345bf98ab86213680ea86cd8f4f44","target":"record","created_at":"2026-05-18T00:34:54Z","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":"e306d61bcea30cc8b1e09d55ed58cbc8d7c16d7361b8cb2ca231c7349151bc81","cross_cats_sorted":["cs.AI","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-09-16T18:59:17Z","title_canon_sha256":"781c4e1336d8dba232598ae9193ef560e0795c1d8b4b9ef09350227480d03a60"},"schema_version":"1.0","source":{"id":"1709.06080","kind":"arxiv","version":1}},"canonical_sha256":"69811140260d0e091755c46c9ad19bae16a25e75fe0a6f66f3acb88bd4663b70","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69811140260d0e091755c46c9ad19bae16a25e75fe0a6f66f3acb88bd4663b70","first_computed_at":"2026-05-18T00:34:54.448938Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:54.448938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LBLo1gaJGFK29BVk9XprNuug1qtut4U2YsMiOGZGZTaCS/A5LoBopqhFzbooC7Z7tGuGOgehNf1V2iTHWRr+AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:54.449612Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.06080","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef1dc74c409b3e11f9f623d31413295fdd2345bf98ab86213680ea86cd8f4f44","sha256:83da323f6393896c902cb772bfe7a11a5e2c5ea82901e7327ceb80fbb4ae9cf1"],"state_sha256":"ca8d254b0de1a3774983280daeef9a08df7e414bd2c38880fc466bcf4617a1a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XOuPcavfmJHrT3xmZRNfJH7qzbHt8EzbXkzr4dOpKkQTeCZM7y0fqb/JIOMEU/BwJX9gziCFP1K7ATn9s73IDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T13:15:01.197536Z","bundle_sha256":"3b7cdbef84acb2cf54e966456c785d0674eb90b57e22f4ec378bb9a5c93bae04"}}