Robust covariance and scatter matrix estimation under Huber’s contamination model
Open accessAbstract
Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber’s $\varepsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.
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The smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.
“the smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Chen et al., 2018
The paper’s Abstract (page unavailable) says that optimizing its matrix-depth function achieves minimax rates under Huber contamination. It does not describe a smoothed-depth estimator or establish that one matches a σα floor in sub-exponential time; the same passage says its high-dimensional depth estimators are challenging to compute. (LLM, conf=0.82)
Experimental — accuracy not yet measuredChen et al. (2018), Theorem 5.1, establish Θ(σ^2(d/N + α^2)) as the squared-error minimax for sub-Gaussian Huber, with no d in the contamination term.
“and indeed Chen et al. (2018), Theorem 5.1, establish Θ(σ 2 (d/N + α 2 )) as the squared-error minimax for sub-Gaussian Huber, with no d in the contamination term.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Chen et al. (2018), Theorem 5.1
Section 5 (Theorem 5.1) gives a general minimax lower bound and illustrates it with a Gaussian location model, obtaining (p/n) ∨ ε²—equivalent in rate to p/n + ε², with no dimension factor on the contamination term. Section 2 (Theorems 2.1 and 2.2) supplies the matching rate for that Gaussian model. The copy does not establish the stated σ²-scaled rate for the broader sub-Gaussian class, and Theorem 5.1 alone does not establish a matching upper bound. (LLM, conf=0.90)
Experimental — accuracy not yet measuredThe minimax-optimal estimator on the breakdown floor is the Tukey halfspace median, whose exact computation is NP-hard for d ≥ 3; the smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.
“The minimax-optimal estimator on the breakdown floor is the Tukey halfspace median (Tukey, 1975;Donoho and Gasko, 1992), whose exact computation is NP-hard for d ≥ 3 (Johnson and Preparata, 1978;Aloupis, 2006); the smoothed-depth estimator of Chen et al. (2018) matches the σα floor in sub-exponential time.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Chen et al. (2018)
Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredChen et al. (2018), Theorem 5.1, prove inf sup E||θ - θ||_2 ≍ σ^2(d/N + α^2), with no d on the squared-error contamination floor.
“Chen et al. (2018), Theorem 5.1, prove inf sup E∥ θ -θ∥ 2 ≍ σ 2 (d/N + α 2 ), with no d on the squared-error contamination floor.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Chen et al. (2018)
Section 5 (Theorem 5.1; page unavailable) derives a minimax lower bound of order (p/n) ∨ ε² for the Gaussian location model with identity covariance. This supports a dimension-free squared-error contamination floor. It does not state the claimed two-sided equivalence for expected loss or the σ² scaling; the upper bound in Section 2 (Theorem 2.1; page unavailable) is a high-probability result under additional conditions. (LLM, conf=0.88)
Experimental — accuracy not yet measured
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Cites
Statistical Estimation and Optimal Recovery
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Research works printed but not linked: 9
- CHEN, M., GAO, C. and REN, Z. (2018). Supplement to "Robust covariance and scatter matrix estimation under Huber's contamination model." DOI:10.1214/17-AOS1607SUPP. · DOI
- DONOHO, D. L. and MONTANARI, A. (2015). Variance breakdown of Huber (M)-estimators: n/p → m ∈ (1, ∞). Preprint. Available at arXiv:1503.02106. · arXiv
- HAN, F. and LIU, H. (2013). Optimal rates of convergence for latent generalized correlation matrix estimation in transelliptical distribution. Preprint. Available at arXiv:1305.6916. · arXiv
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Dataset Security for Machine Learning: Data Poisoning, Backdoor Attacks, and Defenses
Goldblum, M., Tsipras, D., Xie, C., Chen, X., Schwarzschild, A., Song, D., Madry, A., Li, B., & Goldstein, T. (2023) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2022.3162397 · Cited by 362 · see citing works
RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Chen, M., Gao, C., & Ren, Z.
- Affiliations
- University of Pittsburgh (US)
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2018-10-01
- Volume
- 46
- Issue
- 5
- Pages
- 1932-1960
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2963616494
- Cited by
- 146
- References
- 1
- Citations by year
- 2026: 9 · 2025: 22 · 2024: 13 · 2023: 11 · 2022: 20 · 2021: 18 · 2020: 20 · 2019: 16 · 2018: 9 · 2017: 4 · 2016: 3 · 2015: 1
- FWCI
- 12.86
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1712957)
- Yale University
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Random Matrices and Applications
- Keywords
- robust covariance estimation, scatter matrix estimation, minimax optimal rate, outliers
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