{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:HLEKKGYKYRAPVA55ETFU7YTAWD","short_pith_number":"pith:HLEKKGYK","schema_version":"1.0","canonical_sha256":"3ac8a51b0ac440fa83bd24cb4fe260b0c398acdffcce0cbbcab0d7e80faacb6b","source":{"kind":"arxiv","id":"2408.05104","version":4},"attestation_state":"computed","paper":{"title":"Generalised Rank-Constrained Approximations of Hilbert-Schmidt Operators on Separable Hilbert Spaces and Applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.FA","authors_text":"Giuseppe Carere, Han Cheng Lie","submitted_at":"2024-08-09T14:53:12Z","abstract_excerpt":"In this work we solve, for given bounded operators $B,C$ and Hilbert-Schmidt operator $M$ acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, $\\min\\{\\lVert M-BXC\\rVert_{L_2}:\\ \\text{dim ran}\\ X\\leq r\\}.$ This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices $M$, $B$, $C$, $X$, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular va"},"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":"2408.05104","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2024-08-09T14:53:12Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"72960ca483dcd274b9db8622910a664d0caa2227476bec3315a01afe4d396499","abstract_canon_sha256":"fde668f7c7df940ecf6120cc1ae81b0dc4212a23ef25f591dd6ad3ee4322e82d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:05:32.453409Z","signature_b64":"ZKPq564AOkztnA+3aIQOrOdeiansaMKOkJnijcl0hyav9WLIXCEBDBVydVsAJc2LACn+fPFqR5QsmMEeff3OBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ac8a51b0ac440fa83bd24cb4fe260b0c398acdffcce0cbbcab0d7e80faacb6b","last_reissued_at":"2026-05-27T01:05:32.452831Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:05:32.452831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalised Rank-Constrained Approximations of Hilbert-Schmidt Operators on Separable Hilbert Spaces and Applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.FA","authors_text":"Giuseppe Carere, Han Cheng Lie","submitted_at":"2024-08-09T14:53:12Z","abstract_excerpt":"In this work we solve, for given bounded operators $B,C$ and Hilbert-Schmidt operator $M$ acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, $\\min\\{\\lVert M-BXC\\rVert_{L_2}:\\ \\text{dim ran}\\ X\\leq r\\}.$ This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices $M$, $B$, $C$, $X$, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.05104","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.05104/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":"2408.05104","created_at":"2026-05-27T01:05:32.452903+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.05104v4","created_at":"2026-05-27T01:05:32.452903+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.05104","created_at":"2026-05-27T01:05:32.452903+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLEKKGYKYRAP","created_at":"2026-05-27T01:05:32.452903+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLEKKGYKYRAPVA55","created_at":"2026-05-27T01:05:32.452903+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLEKKGYK","created_at":"2026-05-27T01:05:32.452903+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2503.24209","citing_title":"Optimal low-rank posterior mean and distribution approximation in linear Gaussian inverse problems on Hilbert spaces","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD","json":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD.json","graph_json":"https://pith.science/api/pith-number/HLEKKGYKYRAPVA55ETFU7YTAWD/graph.json","events_json":"https://pith.science/api/pith-number/HLEKKGYKYRAPVA55ETFU7YTAWD/events.json","paper":"https://pith.science/paper/HLEKKGYK"},"agent_actions":{"view_html":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD","download_json":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD.json","view_paper":"https://pith.science/paper/HLEKKGYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.05104&json=true","fetch_graph":"https://pith.science/api/pith-number/HLEKKGYKYRAPVA55ETFU7YTAWD/graph.json","fetch_events":"https://pith.science/api/pith-number/HLEKKGYKYRAPVA55ETFU7YTAWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD/action/storage_attestation","attest_author":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD/action/author_attestation","sign_citation":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD/action/citation_signature","submit_replication":"https://pith.science/pith/HLEKKGYKYRAPVA55ETFU7YTAWD/action/replication_record"}},"created_at":"2026-05-27T01:05:32.452903+00:00","updated_at":"2026-05-27T01:05:32.452903+00:00"}