{"paper":{"title":"Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Sums of bounded self-adjoint operators satisfy empirical Bernstein inequalities that depend only on intrinsic dimension","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Aaditya Ramdas, Diego Martinez-Taboada","submitted_at":"2026-05-14T18:00:07Z","abstract_excerpt":"Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori knowledge of the true variance, and often explicitly scale with the ambient dimension, rendering them vacuous for infinite-dimensional or heavily structured operators. Motivated by these challenges, we establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bound"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The random operators are independent, bounded, and compact self-adjoint, and that an empirical estimate of the variance together with the intrinsic dimension can be substituted directly into the concentration bounds without introducing uncontrolled bias or requiring additional oracle information.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives the first empirical Bennett and Bernstein inequalities for bounded compact self-adjoint operators that use intrinsic dimension and empirical variance estimates to achieve dimension-free guarantees.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sums of bounded self-adjoint operators satisfy empirical Bernstein inequalities that depend only on intrinsic dimension","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"47975faaf0900e2aaaaea6f4e13442b35b4cd5e14345b5c50d927718ed56d981"},"source":{"id":"2605.15278","kind":"arxiv","version":1},"verdict":{"id":"4734f0e3-8935-42c2-98fe-5d97c7a2273d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T16:03:58.935745Z","strongest_claim":"We establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces.","one_line_summary":"Derives the first empirical Bennett and Bernstein inequalities for bounded compact self-adjoint operators that use intrinsic dimension and empirical variance estimates to achieve dimension-free guarantees.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The random operators are independent, bounded, and compact self-adjoint, and that an empirical estimate of the variance together with the intrinsic dimension can be substituted directly into the concentration bounds without introducing uncontrolled bias or requiring additional oracle information.","pith_extraction_headline":"Sums of bounded self-adjoint operators satisfy empirical Bernstein inequalities that depend only on intrinsic dimension"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15278/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T16:31:18.382206Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:16:18.213803Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:41:54.253695Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.799246Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"242481518f779412b539bb7971135dc809bde169f129c2f0f32e63d7818923b2"},"references":{"count":13,"sample":[{"doi":"","year":2002,"title":"Ahlswede, R. and Winter, A. (2002). Strong converse for identification via quantum channels.IEEE Transactions on Information Theory, 48(3):569–579. Araki, H. (1975). Relative entropy of states of von ","work_id":"94096b02-bf7b-4062-8257-06b7fca94260","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Bottou, L., Curtis, F","work_id":"9f8336ed-09c9-4f12-8b32-92f6c59d3e06","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"American Mathematical Society. Conway, J. B. (2019).A Course in Functional Analysis, volume","work_id":"8578b47f-1254-4a23-a28f-137028259d03","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Springer. Dauphin, Y. N., Pascanu, R., Gulcehre, C., Cho, K., Ganguli, S., and Bengio, Y. (2014). Identifying and attacking the saddle point problem in high-dimensional non-convex optimization.Advance","work_id":"2c664a92-63ea-48b4-8147-8872d32cad5f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Flammia, S. T., Gross, D., Liu, Y.-K., and Eisert, J. (2012). Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators.New Journal of Physics, 14:095022. Gut","work_id":"7db3ef48-5580-42e4-b931-ace5885e12c8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"3bcd80d252b410e9ba11b5bf203a9b1e471164743e40d1fabcde573dd633e69e","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f567f5d3a8a1e1009a0ecfba772d4a323b03a16a18e6cbf3437f38600a58eb32"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}