{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:MBNTRUP47YKM4NDRBGOSUJ3U64","short_pith_number":"pith:MBNTRUP4","canonical_record":{"source":{"id":"1809.08587","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-09-23T12:32:45Z","cross_cats_sorted":["cs.NE","math.OC","stat.ML"],"title_canon_sha256":"4cea27c5e6772158e1df8f56770a6b94f6767822742538d045e9a4049e21783d","abstract_canon_sha256":"7682a179b6c92b15f437f65dc528628bc78f9e07d0453c9d0c01c6bf609f3fcb"},"schema_version":"1.0"},"canonical_sha256":"605b38d1fcfe14ce3471099d2a2774f73170e5674477e044ce046762fdc2a4bc","source":{"kind":"arxiv","id":"1809.08587","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.08587","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"arxiv_version","alias_value":"1809.08587v4","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08587","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"pith_short_12","alias_value":"MBNTRUP47YKM","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MBNTRUP47YKM4NDR","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MBNTRUP4","created_at":"2026-05-18T12:32:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:MBNTRUP47YKM4NDRBGOSUJ3U64","target":"record","payload":{"canonical_record":{"source":{"id":"1809.08587","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-09-23T12:32:45Z","cross_cats_sorted":["cs.NE","math.OC","stat.ML"],"title_canon_sha256":"4cea27c5e6772158e1df8f56770a6b94f6767822742538d045e9a4049e21783d","abstract_canon_sha256":"7682a179b6c92b15f437f65dc528628bc78f9e07d0453c9d0c01c6bf609f3fcb"},"schema_version":"1.0"},"canonical_sha256":"605b38d1fcfe14ce3471099d2a2774f73170e5674477e044ce046762fdc2a4bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:28.244040Z","signature_b64":"ZGOOOv+UUrCxQrVBVdgoOurTfH0ogCNXLp/iavlsg5nY0ERJwh1TZUkf9DT8VuiZgN2nw1ZFODIrKHRjOQ6dCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"605b38d1fcfe14ce3471099d2a2774f73170e5674477e044ce046762fdc2a4bc","last_reissued_at":"2026-05-17T23:43:28.243609Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:28.243609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.08587","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:43:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xT0OwvOP9IvtiYoCD7nJWhhv5nYNsx7PEYvER9BH2cKTXinCJqzC8AmZd2hNtj5BARZCRlY480l0X4xdlr61Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:10:58.512778Z"},"content_sha256":"33368147540856c29d47dd82056e6e3858e4f24fa3d61e1114a2bbae50fba9cc","schema_version":"1.0","event_id":"sha256:33368147540856c29d47dd82056e6e3858e4f24fa3d61e1114a2bbae50fba9cc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:MBNTRUP47YKM4NDRBGOSUJ3U64","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exponential Convergence Time of Gradient Descent for One-Dimensional Deep Linear Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Ohad Shamir","submitted_at":"2018-09-23T12:32:45Z","abstract_excerpt":"We study the dynamics of gradient descent on objective functions of the form $f(\\prod_{i=1}^{k} w_i)$ (with respect to scalar parameters $w_1,\\ldots,w_k$), which arise in the context of training depth-$k$ linear neural networks. We prove that for standard random initializations, and under mild assumptions on $f$, the number of iterations required for convergence scales exponentially with the depth $k$. We also show empirically that this phenomenon can occur in higher dimensions, where each $w_i$ is a matrix. This highlights a potential obstacle in understanding the convergence of gradient-base"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08587","kind":"arxiv","version":4},"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":"2026-05-17T23:43:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8nilJL20Rs97588zmvAuRcPuPwy/oS76P/4SopJywuCyCofdBm2TaXPwsyDLNkv9kXYgtKxSX0P9qB34GiNYBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:10:58.513407Z"},"content_sha256":"aa0eea94f5d0c3250b61586bdd82096cf75dd2141234c10bd8c335c7649bd8c7","schema_version":"1.0","event_id":"sha256:aa0eea94f5d0c3250b61586bdd82096cf75dd2141234c10bd8c335c7649bd8c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/bundle.json","state_url":"https://pith.science/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/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":"2026-05-25T20:10:58Z","links":{"resolver":"https://pith.science/pith/MBNTRUP47YKM4NDRBGOSUJ3U64","bundle":"https://pith.science/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/bundle.json","state":"https://pith.science/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MBNTRUP47YKM4NDRBGOSUJ3U64/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MBNTRUP47YKM4NDRBGOSUJ3U64","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":"7682a179b6c92b15f437f65dc528628bc78f9e07d0453c9d0c01c6bf609f3fcb","cross_cats_sorted":["cs.NE","math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-09-23T12:32:45Z","title_canon_sha256":"4cea27c5e6772158e1df8f56770a6b94f6767822742538d045e9a4049e21783d"},"schema_version":"1.0","source":{"id":"1809.08587","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.08587","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"arxiv_version","alias_value":"1809.08587v4","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08587","created_at":"2026-05-17T23:43:28Z"},{"alias_kind":"pith_short_12","alias_value":"MBNTRUP47YKM","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MBNTRUP47YKM4NDR","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MBNTRUP4","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:aa0eea94f5d0c3250b61586bdd82096cf75dd2141234c10bd8c335c7649bd8c7","target":"graph","created_at":"2026-05-17T23:43:28Z","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":"We study the dynamics of gradient descent on objective functions of the form $f(\\prod_{i=1}^{k} w_i)$ (with respect to scalar parameters $w_1,\\ldots,w_k$), which arise in the context of training depth-$k$ linear neural networks. We prove that for standard random initializations, and under mild assumptions on $f$, the number of iterations required for convergence scales exponentially with the depth $k$. We also show empirically that this phenomenon can occur in higher dimensions, where each $w_i$ is a matrix. This highlights a potential obstacle in understanding the convergence of gradient-base","authors_text":"Ohad Shamir","cross_cats":["cs.NE","math.OC","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-09-23T12:32:45Z","title":"Exponential Convergence Time of Gradient Descent for One-Dimensional Deep Linear Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08587","kind":"arxiv","version":4},"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":"sha256:33368147540856c29d47dd82056e6e3858e4f24fa3d61e1114a2bbae50fba9cc","target":"record","created_at":"2026-05-17T23:43:28Z","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":"7682a179b6c92b15f437f65dc528628bc78f9e07d0453c9d0c01c6bf609f3fcb","cross_cats_sorted":["cs.NE","math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-09-23T12:32:45Z","title_canon_sha256":"4cea27c5e6772158e1df8f56770a6b94f6767822742538d045e9a4049e21783d"},"schema_version":"1.0","source":{"id":"1809.08587","kind":"arxiv","version":4}},"canonical_sha256":"605b38d1fcfe14ce3471099d2a2774f73170e5674477e044ce046762fdc2a4bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"605b38d1fcfe14ce3471099d2a2774f73170e5674477e044ce046762fdc2a4bc","first_computed_at":"2026-05-17T23:43:28.243609Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:28.243609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZGOOOv+UUrCxQrVBVdgoOurTfH0ogCNXLp/iavlsg5nY0ERJwh1TZUkf9DT8VuiZgN2nw1ZFODIrKHRjOQ6dCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:28.244040Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.08587","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:33368147540856c29d47dd82056e6e3858e4f24fa3d61e1114a2bbae50fba9cc","sha256:aa0eea94f5d0c3250b61586bdd82096cf75dd2141234c10bd8c335c7649bd8c7"],"state_sha256":"e98131f5250d8496bc428d41d33b5dd77197011f93b18da8847221ae92a86e01"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S0CiMKBXs4wVcR3P+z1yutaBwfSiZN0KS1AInJQCG0Og1t592Bur2mwScVoPfshzVtSYPppus03Q4xhMrPHwDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T20:10:58.516728Z","bundle_sha256":"4a116f6c3612bf828e7d2094373d7f3897f3f3b6f490b9f8ebb78a3bb8d82292"}}