{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:G5VSMIE45LKOOS2YFFXHF2KZN5","short_pith_number":"pith:G5VSMIE4","schema_version":"1.0","canonical_sha256":"376b26209cead4e74b58296e72e9596f6d94b722ed5fd81221ff86d37ea73512","source":{"kind":"arxiv","id":"2601.20844","version":3},"attestation_state":"computed","paper":{"title":"$\\mathbb{R}^{2k}$ is Theoretically Large Enough for Embedding-based Top-$k$ Retrieval","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.AI","cs.IR"],"primary_cat":"cs.LG","authors_text":"Ginny Wong, Hanghang Tong, Hang Yin, Lihui Liu, Simon See, Yangqiu Song, Zihao Wang","submitted_at":"2026-01-28T18:45:43Z","abstract_excerpt":"This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison. Our result shows MED is $\\Theta(k)$, independent of $m$, for inner product, Euclidean distance, and cosine similarity. We then consider Robust MED (RMED), where all vectors are unit normed and an $\\epsilon$ gap of scores is required. We derive the $m$-dependent feasibility ceiling $\\epsilon_\\star(m,k)=m/\\sqrt{k(m-1)(m-k)}$, which approaches $1/\\sqrt{k}$ when $m\\gg k$, and a G"},"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":"2601.20844","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.LG","submitted_at":"2026-01-28T18:45:43Z","cross_cats_sorted":["cs.AI","cs.IR"],"title_canon_sha256":"dc7ccf7b70324ac67e8e52b0dca662b988a8a38fcc3f1c457bdd602f35c85451","abstract_canon_sha256":"ac613715cd29d4a3e3cfcc4905cddb29cc596bd5af5c5905bfc9401dbc4d3191"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:09.541589Z","signature_b64":"DsDMeTp6SAqyZDeK3m0QkHDiC8zSAY1jDvJD9Qiv1oU3BHEQIBmZK0PYkicPiDNtL9C/+z0ESE8sVXkBvziPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"376b26209cead4e74b58296e72e9596f6d94b722ed5fd81221ff86d37ea73512","last_reissued_at":"2026-06-03T01:05:09.541113Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:09.541113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\mathbb{R}^{2k}$ is Theoretically Large Enough for Embedding-based Top-$k$ Retrieval","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.AI","cs.IR"],"primary_cat":"cs.LG","authors_text":"Ginny Wong, Hanghang Tong, Hang Yin, Lihui Liu, Simon See, Yangqiu Song, Zihao Wang","submitted_at":"2026-01-28T18:45:43Z","abstract_excerpt":"This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison. Our result shows MED is $\\Theta(k)$, independent of $m$, for inner product, Euclidean distance, and cosine similarity. We then consider Robust MED (RMED), where all vectors are unit normed and an $\\epsilon$ gap of scores is required. We derive the $m$-dependent feasibility ceiling $\\epsilon_\\star(m,k)=m/\\sqrt{k(m-1)(m-k)}$, which approaches $1/\\sqrt{k}$ when $m\\gg k$, and a G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.20844","kind":"arxiv","version":3},"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/2601.20844/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":"2601.20844","created_at":"2026-06-03T01:05:09.541172+00:00"},{"alias_kind":"arxiv_version","alias_value":"2601.20844v3","created_at":"2026-06-03T01:05:09.541172+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.20844","created_at":"2026-06-03T01:05:09.541172+00:00"},{"alias_kind":"pith_short_12","alias_value":"G5VSMIE45LKO","created_at":"2026-06-03T01:05:09.541172+00:00"},{"alias_kind":"pith_short_16","alias_value":"G5VSMIE45LKOOS2Y","created_at":"2026-06-03T01:05:09.541172+00:00"},{"alias_kind":"pith_short_8","alias_value":"G5VSMIE4","created_at":"2026-06-03T01:05:09.541172+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.23556","citing_title":"Is Dimensionality a Barrier for Retrieval Models?","ref_index":59,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5","json":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5.json","graph_json":"https://pith.science/api/pith-number/G5VSMIE45LKOOS2YFFXHF2KZN5/graph.json","events_json":"https://pith.science/api/pith-number/G5VSMIE45LKOOS2YFFXHF2KZN5/events.json","paper":"https://pith.science/paper/G5VSMIE4"},"agent_actions":{"view_html":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5","download_json":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5.json","view_paper":"https://pith.science/paper/G5VSMIE4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2601.20844&json=true","fetch_graph":"https://pith.science/api/pith-number/G5VSMIE45LKOOS2YFFXHF2KZN5/graph.json","fetch_events":"https://pith.science/api/pith-number/G5VSMIE45LKOOS2YFFXHF2KZN5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5/action/storage_attestation","attest_author":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5/action/author_attestation","sign_citation":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5/action/citation_signature","submit_replication":"https://pith.science/pith/G5VSMIE45LKOOS2YFFXHF2KZN5/action/replication_record"}},"created_at":"2026-06-03T01:05:09.541172+00:00","updated_at":"2026-06-03T01:05:09.541172+00:00"}