{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:TOFU5ILFBUP3QUPDT6QIDKKEYH","short_pith_number":"pith:TOFU5ILF","schema_version":"1.0","canonical_sha256":"9b8b4ea1650d1fb851e39fa081a944c1f7076706d89caa522802565ef642fc33","source":{"kind":"arxiv","id":"2607.23017","version":1},"attestation_state":"computed","paper":{"title":"Level-set entropy and sparse randomized embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov","submitted_at":"2026-07-25T03:18:37Z","abstract_excerpt":"Let $\\Pi$ be a $k\\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\\subset{\\mathbb R}^n$, let $U_V:{\\mathbb R}^r\\to{\\mathbb R}^n$ denote an isometry from ${\\mathbb R}^r$ onto $V$. The product $\\Pi U_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $\\Pi U_V$, based on entropy estimates for level sets of vectors $x\\in V$. Combining the method with existing estimates, we show the following. Assume that \\[\n  k"},"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":"2607.23017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-25T03:18:37Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"b8fe9f3e91374a7fa3a1b7c1546f14b7ce93d726ac66828b67d422e84aafcd9b","abstract_canon_sha256":"917ad688864df1fbed5ad8b1b0e5ad301229d328ac662dcd43541da605c56a4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T00:22:07.734402Z","signature_b64":"vNTqTCL93mhD03Rq4O0WtzbEHjx9JUDxDP7ErTavT9DTbGe0WiDBSWnm/mzI6FB1EYItBc1ehJrXKJaE3buUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b8b4ea1650d1fb851e39fa081a944c1f7076706d89caa522802565ef642fc33","last_reissued_at":"2026-07-28T00:22:07.733578Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T00:22:07.733578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Level-set entropy and sparse randomized embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov","submitted_at":"2026-07-25T03:18:37Z","abstract_excerpt":"Let $\\Pi$ be a $k\\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\\subset{\\mathbb R}^n$, let $U_V:{\\mathbb R}^r\\to{\\mathbb R}^n$ denote an isometry from ${\\mathbb R}^r$ onto $V$. The product $\\Pi U_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $\\Pi U_V$, based on entropy estimates for level sets of vectors $x\\in V$. Combining the method with existing estimates, we show the following. Assume that \\[\n  k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23017","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/2607.23017/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":"2607.23017","created_at":"2026-07-28T00:22:07.734000+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.23017v1","created_at":"2026-07-28T00:22:07.734000+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23017","created_at":"2026-07-28T00:22:07.734000+00:00"},{"alias_kind":"pith_short_12","alias_value":"TOFU5ILFBUP3","created_at":"2026-07-28T00:22:07.734000+00:00"},{"alias_kind":"pith_short_16","alias_value":"TOFU5ILFBUP3QUPD","created_at":"2026-07-28T00:22:07.734000+00:00"},{"alias_kind":"pith_short_8","alias_value":"TOFU5ILF","created_at":"2026-07-28T00:22:07.734000+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/TOFU5ILFBUP3QUPDT6QIDKKEYH","json":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH.json","graph_json":"https://pith.science/api/pith-number/TOFU5ILFBUP3QUPDT6QIDKKEYH/graph.json","events_json":"https://pith.science/api/pith-number/TOFU5ILFBUP3QUPDT6QIDKKEYH/events.json","paper":"https://pith.science/paper/TOFU5ILF"},"agent_actions":{"view_html":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH","download_json":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH.json","view_paper":"https://pith.science/paper/TOFU5ILF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.23017&json=true","fetch_graph":"https://pith.science/api/pith-number/TOFU5ILFBUP3QUPDT6QIDKKEYH/graph.json","fetch_events":"https://pith.science/api/pith-number/TOFU5ILFBUP3QUPDT6QIDKKEYH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH/action/storage_attestation","attest_author":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH/action/author_attestation","sign_citation":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH/action/citation_signature","submit_replication":"https://pith.science/pith/TOFU5ILFBUP3QUPDT6QIDKKEYH/action/replication_record"}},"created_at":"2026-07-28T00:22:07.734000+00:00","updated_at":"2026-07-28T00:22:07.734000+00:00"}