Excess-Risk Consistency of Group-hard Thresholding Estimator in Robust Estimation of Gaussian Mean

Arshak, Minasyan,

doi:10.3103/s1068362320030073

Cited by 1 publication

(1 citation statement)

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“…Another interesting avenue for future research is to find an estimator that is rate-optimal, computationally tractable, with breakdown point equal to 1/2 and, in the same time, asymptotically optimal in the sense that its risk is of order p/n when ε tends to zero. On a related note, it would be interesting to push further the exploration of second-order properties of the risk started in (Minasyan, 2020). Finally, an open question is how the minimax risk blows-up when the contamination rate tends to ε.…”

Section: Breakdown Pointmentioning

confidence: 99%

Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction

Bateni¹,

Arshak²,

Dalalyan³

2022

Preprint

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We study the problem of robust estimation of the mean vector of a sub-Gaussian distribution. We introduce an estimator based on spectral dimension reduction (SDR) and establish a finite sample upper bound on its error that is minimax-optimal up to a logarithmic factor. Furthermore, we prove that the breakdown point of the SDR estimator is equal to 1/2, the highest possible value of the breakdown point. In addition, the SDR estimator is equivariant by similarity transforms and has low computational complexity. More precisely, in the case of n vectors of dimension p-at most εn out of which are adversarially corrupted-the SDR estimator has a squared error of order (rΣ/n + ε 2 log(1/ε))log p and a running time of order p 3 + np 2 . Here, r Σ ≤ p is the effective rank of the covariance matrix of the reference distribution. Another advantage of the SDR estimator is that it does not require knowledge of the contamination rate and does not involve sample splitting. We also investigate extensions of the proposed algorithm and of the obtained results in the case of (partially) unknown covariance matrix.

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Section: Breakdown Pointmentioning

confidence: 99%