{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:32UK2ZWI5AOEVO5BZHZCLHZWZL","short_pith_number":"pith:32UK2ZWI","canonical_record":{"source":{"id":"1812.04754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-12-12T00:36:17Z","cross_cats_sorted":["cs.AI","stat.ML"],"title_canon_sha256":"a214d12d3469fdb156efe9ea2c61085c1da72854b16a2c0887949b33f7fca751","abstract_canon_sha256":"20fc5fdbf0fbf77b2523dc57e260e5db703176536904ef9f972065d46f592ac9"},"schema_version":"1.0"},"canonical_sha256":"dea8ad66c8e81c4abba1c9f2259f36cae91c404c739f97c31fbc16bffefa2c78","source":{"kind":"arxiv","id":"1812.04754","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04754","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04754v1","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04754","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"pith_short_12","alias_value":"32UK2ZWI5AOE","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"32UK2ZWI5AOEVO5B","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"32UK2ZWI","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:32UK2ZWI5AOEVO5BZHZCLHZWZL","target":"record","payload":{"canonical_record":{"source":{"id":"1812.04754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-12-12T00:36:17Z","cross_cats_sorted":["cs.AI","stat.ML"],"title_canon_sha256":"a214d12d3469fdb156efe9ea2c61085c1da72854b16a2c0887949b33f7fca751","abstract_canon_sha256":"20fc5fdbf0fbf77b2523dc57e260e5db703176536904ef9f972065d46f592ac9"},"schema_version":"1.0"},"canonical_sha256":"dea8ad66c8e81c4abba1c9f2259f36cae91c404c739f97c31fbc16bffefa2c78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:29.580571Z","signature_b64":"zFCZfmx+L+zepHgcV2tccpAukLFo+luwelcnzXM2pskucGJHEIhK/NUnfEXIiiV+2nJX6P9FkUYGhadqpgxjBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dea8ad66c8e81c4abba1c9f2259f36cae91c404c739f97c31fbc16bffefa2c78","last_reissued_at":"2026-05-17T23:58:29.579846Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:29.579846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.04754","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":"2026-05-17T23:58:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4zHmL3d3iTEe+EDnnRfoaUB7eU45fVjqTqzRc4NPwVKrtF2QSdhpfJzMlnnP0uZPOHwYfSBCmGl6oHNxRK++AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T00:41:24.291555Z"},"content_sha256":"a6913fb7b8d07b7b256d9e6426c0a0cc93096786fa8d998e291f39cfe83dd508","schema_version":"1.0","event_id":"sha256:a6913fb7b8d07b7b256d9e6426c0a0cc93096786fa8d998e291f39cfe83dd508"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:32UK2ZWI5AOEVO5BZHZCLHZWZL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gradient Descent Happens in a Tiny Subspace","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","stat.ML"],"primary_cat":"cs.LG","authors_text":"Daniel A. Roberts, Ethan Dyer, Guy Gur-Ari","submitted_at":"2018-12-12T00:36:17Z","abstract_excerpt":"We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04754","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":"2026-05-17T23:58:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GArZl0iv1YFfd5aBdm7oL+LnNrpwmYIHKU5yDgcItZ1UowHcJDmxQpDUlNL98Cn5IgkQarf5nPHo5MGB44jSDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T00:41:24.291902Z"},"content_sha256":"a6348877694433fda2e41605cb821887be2573968453d869057ef75ca311b74b","schema_version":"1.0","event_id":"sha256:a6348877694433fda2e41605cb821887be2573968453d869057ef75ca311b74b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/bundle.json","state_url":"https://pith.science/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/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":"2026-06-02T00:41:24Z","links":{"resolver":"https://pith.science/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL","bundle":"https://pith.science/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/bundle.json","state":"https://pith.science/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/32UK2ZWI5AOEVO5BZHZCLHZWZL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:32UK2ZWI5AOEVO5BZHZCLHZWZL","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":"20fc5fdbf0fbf77b2523dc57e260e5db703176536904ef9f972065d46f592ac9","cross_cats_sorted":["cs.AI","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-12-12T00:36:17Z","title_canon_sha256":"a214d12d3469fdb156efe9ea2c61085c1da72854b16a2c0887949b33f7fca751"},"schema_version":"1.0","source":{"id":"1812.04754","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04754","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04754v1","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04754","created_at":"2026-05-17T23:58:29Z"},{"alias_kind":"pith_short_12","alias_value":"32UK2ZWI5AOE","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"32UK2ZWI5AOEVO5B","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"32UK2ZWI","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:a6348877694433fda2e41605cb821887be2573968453d869057ef75ca311b74b","target":"graph","created_at":"2026-05-17T23:58:29Z","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 show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.","authors_text":"Daniel A. Roberts, Ethan Dyer, Guy Gur-Ari","cross_cats":["cs.AI","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-12-12T00:36:17Z","title":"Gradient Descent Happens in a Tiny Subspace"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04754","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:a6913fb7b8d07b7b256d9e6426c0a0cc93096786fa8d998e291f39cfe83dd508","target":"record","created_at":"2026-05-17T23:58:29Z","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":"20fc5fdbf0fbf77b2523dc57e260e5db703176536904ef9f972065d46f592ac9","cross_cats_sorted":["cs.AI","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-12-12T00:36:17Z","title_canon_sha256":"a214d12d3469fdb156efe9ea2c61085c1da72854b16a2c0887949b33f7fca751"},"schema_version":"1.0","source":{"id":"1812.04754","kind":"arxiv","version":1}},"canonical_sha256":"dea8ad66c8e81c4abba1c9f2259f36cae91c404c739f97c31fbc16bffefa2c78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dea8ad66c8e81c4abba1c9f2259f36cae91c404c739f97c31fbc16bffefa2c78","first_computed_at":"2026-05-17T23:58:29.579846Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:29.579846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zFCZfmx+L+zepHgcV2tccpAukLFo+luwelcnzXM2pskucGJHEIhK/NUnfEXIiiV+2nJX6P9FkUYGhadqpgxjBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:29.580571Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.04754","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a6913fb7b8d07b7b256d9e6426c0a0cc93096786fa8d998e291f39cfe83dd508","sha256:a6348877694433fda2e41605cb821887be2573968453d869057ef75ca311b74b"],"state_sha256":"a5230a63ad987fd126b508b92df25677176d2924b60993225c7ea63fe3180687"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oleo0ZKphSFD93jBo9/lKkQcVgqE4h+GwlB3BqdhlFs5oLao7S+Te1NpvwXtWZcNnT2j/fGvMDg0iJQaXBRCAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T00:41:24.293873Z","bundle_sha256":"b235e4a978246a6f8d019884d191bb80817e1d642113380ff26b487b2b5d0338"}}