Adaptive estimation of the sparsity in the Gaussian vector model

Carpentier, Alexandra; Verzélen, Nicolas

doi:10.48550/arxiv.1703.00167

Cited by 3 publications

(30 citation statements)

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“…The Gaussian sequence model Y = θ * +σǫ corresponds to case p = n and X = I p . In [18], we have pinpointed the minimax separation distances for all k 0 and ∆ both when σ is known and σ is unknown. In particular, the optimal separation distance actually depends on the size k 0 of the null hypothesis for large k 0 but is significantly smaller than what is obtained by infimum tests strategies such as those in [26].…”

Section: Previous Results and Related Literaturementioning

confidence: 99%

“…Generally speaking, [18] is closely related to the aims and results of this paper, but there is a significant challenge in adapting the results in [18] which are available for the Gaussian sequence setting, to the linear regression setting.…”

Section: Previous Results and Related Literaturementioning

confidence: 99%

“…Related to this problem, some authors [11,12,33,34] have considered the problem of estimating θ * 0 in the Gaussian sequence model in a Bayesian framework where all θ i 's are sampled according to some mixture distribution. Although some of the ideas can be borrowed from their work, this Bayesian setting is quite different (see [18] for a discussion).…”

Section: Previous Results and Related Literaturementioning

confidence: 99%

“…• For larger k 0 ∈ [p 1/2−ζ , cn/ log(p)] (where ζ > 0 is an arbitrarily small absolute constant, and where c > 0 is an absolute constant), then both upper and lower bounds are new. The lower bound is based on moment matching strategies and best polynomial approximation akin to those of [18] in the Gaussian model. But the derivation is significantly more involved in the regression setting.…”

Section: Our Resultsmentioning

confidence: 99%

“…In fact, it is most challenging to prove the minimax lower bound in the regime ∆ > k 0 > √ p as we cannot do any reduction to signal detection problem and we need to take into account that both the null and the alternative hypotheses are composite. As for the Gaussian sequence model [18], we use a general moment matching technique [38], but the non-orthogonal design matrix X makes the computations more tricky.…”

Section: Minimax Lower Boundmentioning

confidence: 99%

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Optimal Sparsity Testing in Linear regression Model

Carpentier¹,

Verzélen²

2019

Preprint

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We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression parameter θ * is less than or equal to k 0 . We pinpoint the minimax separation distances for this problem, which amounts to quantifying how far a k 1 -sparse vector θ * has to be from the set of k 0 -sparse vectors so that a test is able to reject the null hypothesis with high probability. Two scenarios are considered. In the independent scenario, the covariates are i.i.d. normally distributed and the noise level is known. In the general scenario, both the covariance matrix of the covariates and the noise level are unknown. Although the minimax separation distances differ in these two scenarios, both of them actually depend on k 0 and k 1 illustrating that for this composite-composite testing problem both the size of the null and of the alternative hypotheses play a key role. Along the way, we introduce a new variable selection procedure, which can be of independent interest. Property P1. A test φ satisfies (P1[α]) if its type I error probability is less than or equal to α, that is sup θ∈B 0) on a set Θ if its type II error probability is uniformly less than or equal to β, uniformly on Θ, that is inf θ∈Θ P θ,σ [φ = 1] ≥ 1 − β

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Section: Previous Results and Related Literaturementioning

confidence: 99%

Section: Previous Results and Related Literaturementioning

confidence: 99%

Section: Previous Results and Related Literaturementioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Minimax Lower Boundmentioning

confidence: 99%

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Optimal Sparsity Testing in Linear regression Model

Carpentier¹,

Verzélen²

2019

Preprint

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On estimation of nonsmooth functionals of sparse normal means

Collier¹,

Comminges²,

Tsybakov³

2020

Bernoulli

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We study the problem of estimation of Nγ (θ) = d i=1 |θi| γ for γ > 0 and of the ℓγ -norm of θ for γ ≥ 1 based on the observations yi = θi +εξi, i = 1, . . . , d, where θ = (θ1, . . . , θ d ) are unknown parameters, ε > 0 is known, and ξi are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of s-sparse vectors θ and we propose estimators achieving this rate.

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Estimating minimum effect with outlier selection

Carpentier,

Delattre,

Roquain

et al. 2018

Preprint

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We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular structural assumptions on the distribution of the contaminated samples. As for the problem of selecting the outliers, we formulate the problem in a multiple testing framework for which the location/scaling of the null hypotheses are unknown. We rigorously prove how estimating the null hypothesis is possible while maintaining a theoretical guarantee on the amount of the falsely selected outliers, both through false discovery rate (FDR) or post hoc bounds. As a by-product, we address a long-standing open issue on FDR control under equi-correlation, which reinforces the interest of removing dependency when making multiple testing.

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Adaptive estimation of the sparsity in the Gaussian vector model

Cited by 3 publications

References 0 publications

Optimal Sparsity Testing in Linear regression Model

Optimal Sparsity Testing in Linear regression Model

On estimation of nonsmooth functionals of sparse normal means

Estimating minimum effect with outlier selection

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