Adaptive robust estimation in sparse vector model

Comminges, Laëtitia; Collier, Olivier; Ndaoud, Mohamed; Tsybakov, Alexandre B.

doi:10.1214/20-aos2002

Cited by 8 publications

(5 citation statements)

References 23 publications

(45 reference statements)

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“…Sample is a conservative estimator of σ 2 , and characterize its exact bias. We first introduce an auxiliary result that specifies the rate of convergence for a median-based estimator of the variance in the sparse vector model (Comminges et al, 2021). For a vector θ ∈ R n , we use θ 0 to denote its 0 norm, i.e.…”

Section: A5 Proof Of Proposition 4 and Computation Of P σSelectivementioning

confidence: 99%

Selective inference for k-means clustering

Chen¹,

Witten²

2022

Preprint

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We consider the problem of testing for a difference in means between clusters of observations identified via k-means clustering. In this setting, classical hypothesis tests lead to an inflated Type I error rate. To overcome this problem, we take a selective inference approach. We propose a finite-sample p-value that controls the selective Type I error for a test of the difference in means between a pair of clusters obtained using k-means clustering, and show that it can be efficiently computed. We apply our proposal in simulation, and on hand-written digits data and single-cell RNA-sequencing data.

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Section: A5 Proof Of Proposition 4 and Computation Of P σSelectivementioning

confidence: 99%

Selective inference for k-means clustering

Chen¹,

Witten²

2022

Preprint

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“…On the one hand, if the value of γ is at most of order (r Σ /n) log(1/ε) + ε log(1/ε) then Theorem 3 implies that the estimation error is of the same order as in the case of known covariance matrix Σ (Theorem 1). For instance, if the matrix Σ is assumed to be diagonal, one can defined Σ as the diagonal matrix composed of robust estimators of the variances of univariate contaminated Gaussian samples; see, for instance, Section 2 in (Comminges et al, 2021). For recent advances on robust estimation of (non-diagonal) covariance matrices by computationally tractable algorithms we refer the reader to (Cheng et al, 2019b).…”

Section: 2mentioning

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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“…The proof of the lower bound in inspired from [CCNT21] where the goal was to estimate the nuisance parameter θ rather than the signal itself β .…”

Section: Main Ideas Of the Proofmentioning

confidence: 99%

“…Notice that θ is not exactly of sparsity less than o but we will deal with this technicality exactly as in [CCNT21]. Finally, set…”

Section: Main Ideas Of the Proofmentioning

confidence: 99%

Robust and Tuning-Free Sparse Linear Regression via Square-Root Slope

Minsker¹,

Ndaoud²,

Wang³

2022

Preprint

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We consider the high-dimensional linear regression model and assume that a fraction of the responses are contaminated by an adversary with complete knowledge of the data and the underlying distribution. We are interested in the situation when the dense additive noise can be heavy-tailed but the predictors have sub-Gaussian distribution. We establish minimax lower bounds that depend on the the fraction of the contaminated data and the tails of the additive noise. Moreover, we design a modification of the square root Slope estimator with several desirable features: (a) it is provably robust to adversarial contamination, with the performance guarantees that take the form of sub-Gaussian deviation inequalities and match the lower error bounds up to log-factors; (b) it is fully adaptive with respect to the unknown sparsity level and the variance of the noise, and (c) it is computationally tractable as a solution of a convex optimization problem. To analyze the performance of the proposed estimator, we prove several properties of matrices with sub-Gaussian rows that could be of independent interest.

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Adaptive robust estimation in sparse vector model

Cited by 8 publications

References 23 publications

Selective inference for k-means clustering

Selective inference for k-means clustering

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

Robust and Tuning-Free Sparse Linear Regression via Square-Root Slope

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