{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:U76PY55VW2IPKGUNCKHCN4MRVP","short_pith_number":"pith:U76PY55V","schema_version":"1.0","canonical_sha256":"a7fcfc77b5b690f51a8d128e26f191abe3f69ad5d6e1e3f1f46dde4832fd14a1","source":{"kind":"arxiv","id":"2606.22406","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic Signal Subspace Recovery in Softmax Attention Models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT","stat.ML"],"primary_cat":"cs.LG","authors_text":"Lan V. Truong","submitted_at":"2026-06-21T09:38:59Z","abstract_excerpt":"Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynami"},"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":"2606.22406","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-21T09:38:59Z","cross_cats_sorted":["cs.IT","math.IT","stat.ML"],"title_canon_sha256":"ca9796593bf713ce00b97df9750a218bc6567dcc96d5fe3709865ab2007eb390","abstract_canon_sha256":"51b2e45aae3c7602663cb27552a74121745c01bf2d993157884c8db4927acf68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:37.550991Z","signature_b64":"GrCIt1k1r69vjqDxDqNIphxt5Uunend5z5FkWF52LuUk9OwNnlcSo5nvxIB/I32hMfDxbkT3VG+xoh7DBm1NAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7fcfc77b5b690f51a8d128e26f191abe3f69ad5d6e1e3f1f46dde4832fd14a1","last_reissued_at":"2026-06-23T02:13:37.550647Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:37.550647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic Signal Subspace Recovery in Softmax Attention Models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT","stat.ML"],"primary_cat":"cs.LG","authors_text":"Lan V. Truong","submitted_at":"2026-06-21T09:38:59Z","abstract_excerpt":"Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynami"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22406","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/2606.22406/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":"2606.22406","created_at":"2026-06-23T02:13:37.550705+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.22406v1","created_at":"2026-06-23T02:13:37.550705+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22406","created_at":"2026-06-23T02:13:37.550705+00:00"},{"alias_kind":"pith_short_12","alias_value":"U76PY55VW2IP","created_at":"2026-06-23T02:13:37.550705+00:00"},{"alias_kind":"pith_short_16","alias_value":"U76PY55VW2IPKGUN","created_at":"2026-06-23T02:13:37.550705+00:00"},{"alias_kind":"pith_short_8","alias_value":"U76PY55V","created_at":"2026-06-23T02:13:37.550705+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP","json":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP.json","graph_json":"https://pith.science/api/pith-number/U76PY55VW2IPKGUNCKHCN4MRVP/graph.json","events_json":"https://pith.science/api/pith-number/U76PY55VW2IPKGUNCKHCN4MRVP/events.json","paper":"https://pith.science/paper/U76PY55V"},"agent_actions":{"view_html":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP","download_json":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP.json","view_paper":"https://pith.science/paper/U76PY55V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.22406&json=true","fetch_graph":"https://pith.science/api/pith-number/U76PY55VW2IPKGUNCKHCN4MRVP/graph.json","fetch_events":"https://pith.science/api/pith-number/U76PY55VW2IPKGUNCKHCN4MRVP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP/action/storage_attestation","attest_author":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP/action/author_attestation","sign_citation":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP/action/citation_signature","submit_replication":"https://pith.science/pith/U76PY55VW2IPKGUNCKHCN4MRVP/action/replication_record"}},"created_at":"2026-06-23T02:13:37.550705+00:00","updated_at":"2026-06-23T02:13:37.550705+00:00"}