{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:KGINJC3CUXJ7N24WGEDAZ5FLH2","short_pith_number":"pith:KGINJC3C","schema_version":"1.0","canonical_sha256":"5190d48b62a5d3f6eb9631060cf4ab3e93d6b8660557b06a8c46208b1d01e2ed","source":{"kind":"arxiv","id":"2209.14863","version":2},"attestation_state":"computed","paper":{"title":"Neural Networks Efficiently Learn Low-Dimensional Representations with SGD","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Alireza Mousavi-Hosseini, Ioannis Mitliagkas, Manuela Girotti, Murat A. Erdogdu, Sejun Park","submitted_at":"2022-09-29T15:29:10Z","abstract_excerpt":"We study the problem of training a two-layer neural network (NN) of arbitrary width using stochastic gradient descent (SGD) where the input $\\boldsymbol{x}\\in \\mathbb{R}^d$ is Gaussian and the target $y \\in \\mathbb{R}$ follows a multiple-index model, i.e., $y=g(\\langle\\boldsymbol{u_1},\\boldsymbol{x}\\rangle,...,\\langle\\boldsymbol{u_k},\\boldsymbol{x}\\rangle)$ with a noisy link function $g$. We prove that the first-layer weights of the NN converge to the $k$-dimensional principal subspace spanned by the vectors $\\boldsymbol{u_1},...,\\boldsymbol{u_k}$ of the true model, when online SGD with weight"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2209.14863","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2022-09-29T15:29:10Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"1ede5c83aa681bf76d72dc29ce267d04536e8cd2dc94262e247b23a6bb8e0e5a","abstract_canon_sha256":"151afac87c87573a35a8bb06ffa94cbb8658ef21f62f62b68f4f81f05142a934"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:51:41.978744Z","signature_b64":"brhkRL2Z67eUWqvYvqk5VEzB3ve/OpXCYHrQb9zM2nDbFZSCFeNFWVpf5h2iFNDQWej3pAmZyrWeFzINURRwCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5190d48b62a5d3f6eb9631060cf4ab3e93d6b8660557b06a8c46208b1d01e2ed","last_reissued_at":"2026-07-05T05:51:41.978300Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:51:41.978300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Neural Networks Efficiently Learn Low-Dimensional Representations with SGD","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Alireza Mousavi-Hosseini, Ioannis Mitliagkas, Manuela Girotti, Murat A. Erdogdu, Sejun Park","submitted_at":"2022-09-29T15:29:10Z","abstract_excerpt":"We study the problem of training a two-layer neural network (NN) of arbitrary width using stochastic gradient descent (SGD) where the input $\\boldsymbol{x}\\in \\mathbb{R}^d$ is Gaussian and the target $y \\in \\mathbb{R}$ follows a multiple-index model, i.e., $y=g(\\langle\\boldsymbol{u_1},\\boldsymbol{x}\\rangle,...,\\langle\\boldsymbol{u_k},\\boldsymbol{x}\\rangle)$ with a noisy link function $g$. We prove that the first-layer weights of the NN converge to the $k$-dimensional principal subspace spanned by the vectors $\\boldsymbol{u_1},...,\\boldsymbol{u_k}$ of the true model, when online SGD with weight"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.14863","kind":"arxiv","version":2},"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/2209.14863/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2209.14863","created_at":"2026-07-05T05:51:41.978370+00:00"},{"alias_kind":"arxiv_version","alias_value":"2209.14863v2","created_at":"2026-07-05T05:51:41.978370+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.14863","created_at":"2026-07-05T05:51:41.978370+00:00"},{"alias_kind":"pith_short_12","alias_value":"KGINJC3CUXJ7","created_at":"2026-07-05T05:51:41.978370+00:00"},{"alias_kind":"pith_short_16","alias_value":"KGINJC3CUXJ7N24W","created_at":"2026-07-05T05:51:41.978370+00:00"},{"alias_kind":"pith_short_8","alias_value":"KGINJC3C","created_at":"2026-07-05T05:51:41.978370+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2512.14473","citing_title":"Sharp convergence rates for Spectral methods via the feature space decomposition method","ref_index":17,"is_internal_anchor":false},{"citing_arxiv_id":"2605.20235","citing_title":"Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine","ref_index":39,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2","json":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2.json","graph_json":"https://pith.science/api/pith-number/KGINJC3CUXJ7N24WGEDAZ5FLH2/graph.json","events_json":"https://pith.science/api/pith-number/KGINJC3CUXJ7N24WGEDAZ5FLH2/events.json","paper":"https://pith.science/paper/KGINJC3C"},"agent_actions":{"view_html":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2","download_json":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2.json","view_paper":"https://pith.science/paper/KGINJC3C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2209.14863&json=true","fetch_graph":"https://pith.science/api/pith-number/KGINJC3CUXJ7N24WGEDAZ5FLH2/graph.json","fetch_events":"https://pith.science/api/pith-number/KGINJC3CUXJ7N24WGEDAZ5FLH2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2/action/storage_attestation","attest_author":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2/action/author_attestation","sign_citation":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2/action/citation_signature","submit_replication":"https://pith.science/pith/KGINJC3CUXJ7N24WGEDAZ5FLH2/action/replication_record"}},"created_at":"2026-07-05T05:51:41.978370+00:00","updated_at":"2026-07-05T05:51:41.978370+00:00"}