Statistical Guarantees for Covariance Estimation with Adversarial Corruption under Constraints
Abstract
We obtain a near-minimax rate on covariance estimation of adversarially corrupted Gaussian data in Frobenius norm when the covariance estimation is assumed to lie in a star-shaped set K ⊆ ℝn × n. Assuming a known upper bound of ε < 1/32 on the fraction of the N observations of data X are arbitrarily corrupted by an omniscient adversary, we obtain the near-minimax rate (up to constants) of (ε2 ∨ δ⋆2 )∧ d2 ≲ infΣ̂ supΣ∈ K supC 𝔼 | Σ̂(C(X)) − Σ|F2 ≲ (ε2log2(1/ε) ∨ δ⋆2 )∧ d2 where d is the diameter of the set K, N ≳ supδ> 0 log Mloc(δ, 2c), and δ*=sup{ δ≥ 0: N/Υ2 δ2≤ log MKloc(δ,c) } for an arbitrary large constant c and the local metric entropy of K, MlocK(δ,c). We believe this is the first such general result on structured robust covariance estimation in the Frobenius norm.
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BibTeX
@misc{ozminkowski2026statistical,
title = {{Statistical Guarantees for Covariance Estimation with Adversarial Corruption under Constraints}},
author = {Samuel Ozminkowski and Hongmei Jiang and Matey Neykov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10840},
eprint = {2610.10840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10840},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Statistical Guarantees for Covariance Estimation with Adversarial Corruption under Constraints
AU - Samuel Ozminkowski
AU - Hongmei Jiang
AU - Matey Neykov
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10840
UR - https://arxiv.org/abs/2610.10840
AB - We obtain a near-minimax rate on covariance estimation of adversarially corrupted Gaussian data in Frobenius norm when the covariance estimation is assumed to lie in a star-shaped set K ⊆ ℝ^n × n. Assuming a known upper bound of ε < 1/32 on the fraction of the N observations of data X are arbitrarily corrupted by an omniscient adversary, we obtain the near-minimax rate (up to constants) of (ε^2 ∨ δ^⋆2 )∧ d^2 ≲ inf_Σ̂ sup_Σ∈ K sup_C 𝔼 | Σ̂(C(X)) − Σ|_F^2 ≲ (ε^2log^2(1/ε) ∨ δ^⋆2 )∧ d^2 where d is the diameter of the set K, N ≳ sup_δ> 0 log M^loc(δ, 2c), and δ^*=sup{ δ≥ 0: N/Υ^2 δ^2≤ log M_K^loc(δ,c) } for an arbitrary large constant c and the local metric entropy of K, M^loc_K(δ,c). We believe this is the first such general result on structured robust covariance estimation in the Frobenius norm.
ER -