{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:3HTNBT3XPLQJQRGF5LSCNCBI52","short_pith_number":"pith:3HTNBT3X","schema_version":"1.0","canonical_sha256":"d9e6d0cf777ae09844c5eae4268828eeb44070d9babcfde3885cb7d30d012bb2","source":{"kind":"arxiv","id":"1409.1403","version":1},"attestation_state":"computed","paper":{"title":"Nonlinear tensor product approximation of functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"stat.ML","authors_text":"D. Bazarkhanov, V. Temlyakov","submitted_at":"2014-09-04T11:12:48Z","abstract_excerpt":"We are interested in approximation of a multivariate function $f(x_1,\\dots,x_d)$ by linear combinations of products $u^1(x_1)\\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel $f(x_1,x_2)$. There are known results on the rate of decay of errors of best bilinear approximation "},"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":"1409.1403","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2014-09-04T11:12:48Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"2dcc3e3e0f2ebf9ea9bb1f6645df932f9433bd42f040479841241f9726ea27e3","abstract_canon_sha256":"ff74d75e89dab12d49fb09e1cd3cb78d093f0a8bce89cac5bae62277fa8a953a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:27.056547Z","signature_b64":"6q6eCTDy1fDRVsVT0R8z1Vx8h4uRAKsrjp4eNuViPSvNDVjbZDhxt76hkHrMnyQ46Sq65K/g1jJKBLoXxYvhCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9e6d0cf777ae09844c5eae4268828eeb44070d9babcfde3885cb7d30d012bb2","last_reissued_at":"2026-05-18T02:43:27.055987Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:27.055987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonlinear tensor product approximation of functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"stat.ML","authors_text":"D. Bazarkhanov, V. Temlyakov","submitted_at":"2014-09-04T11:12:48Z","abstract_excerpt":"We are interested in approximation of a multivariate function $f(x_1,\\dots,x_d)$ by linear combinations of products $u^1(x_1)\\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel $f(x_1,x_2)$. There are known results on the rate of decay of errors of best bilinear approximation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1403","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1409.1403","created_at":"2026-05-18T02:43:27.056073+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.1403v1","created_at":"2026-05-18T02:43:27.056073+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1403","created_at":"2026-05-18T02:43:27.056073+00:00"},{"alias_kind":"pith_short_12","alias_value":"3HTNBT3XPLQJ","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"3HTNBT3XPLQJQRGF","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"3HTNBT3X","created_at":"2026-05-18T12:28:11.866339+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/3HTNBT3XPLQJQRGF5LSCNCBI52","json":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52.json","graph_json":"https://pith.science/api/pith-number/3HTNBT3XPLQJQRGF5LSCNCBI52/graph.json","events_json":"https://pith.science/api/pith-number/3HTNBT3XPLQJQRGF5LSCNCBI52/events.json","paper":"https://pith.science/paper/3HTNBT3X"},"agent_actions":{"view_html":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52","download_json":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52.json","view_paper":"https://pith.science/paper/3HTNBT3X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.1403&json=true","fetch_graph":"https://pith.science/api/pith-number/3HTNBT3XPLQJQRGF5LSCNCBI52/graph.json","fetch_events":"https://pith.science/api/pith-number/3HTNBT3XPLQJQRGF5LSCNCBI52/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52/action/storage_attestation","attest_author":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52/action/author_attestation","sign_citation":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52/action/citation_signature","submit_replication":"https://pith.science/pith/3HTNBT3XPLQJQRGF5LSCNCBI52/action/replication_record"}},"created_at":"2026-05-18T02:43:27.056073+00:00","updated_at":"2026-05-18T02:43:27.056073+00:00"}