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pith:ZXJB54JV

pith:2026:ZXJB54JVS3OZYCCKSC3Y5ISW2C

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Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators

Aaditya Ramdas, Diego Martinez-Taboada

Sums of bounded self-adjoint operators satisfy empirical Bernstein inequalities that depend only on intrinsic dimension

arxiv:2605.15278 v1 · 2026-05-14 · math.ST · stat.TH

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

13 extracted · 13 resolved · 1 Pith anchors

[1] 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 2002

[2] Bottou, L., Curtis, F 2018

[3] American Mathematical Society. Conway, J. B. (2019).A Course in Functional Analysis, volume 2019

[4] 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 2014

[5] 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 2012

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:50.382456Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cdd21ef13596dd9c084a90b78ea256d0805a33baec8dab18f737c36e303ffafd

Aliases

arxiv: 2605.15278 · arxiv_version: 2605.15278v1 · doi: 10.48550/arxiv.2605.15278 · pith_short_12: ZXJB54JVS3OZ · pith_short_16: ZXJB54JVS3OZYCCK · pith_short_8: ZXJB54JV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZXJB54JVS3OZYCCKSC3Y5ISW2C \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: cdd21ef13596dd9c084a90b78ea256d0805a33baec8dab18f737c36e303ffafd

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "b36935a4cc3e82c269db5dd10d72d077814e6beba00c7fa64c1f5158341414e0",
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    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.ST",
    "submitted_at": "2026-05-14T18:00:07Z",
    "title_canon_sha256": "2d0eefe4e3c7ac99fc5eadc4744159def9776f16e05b0a8ac3817abddee86631"
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