Estimation and confidence sets for sparse normal mixtures

Cai, Tommaso; Jin, Jiashun; Low, Mark G.

doi:10.1214/009053607000000334

Cited by 79 publications

(129 citation statements)

References 13 publications

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“…For the Gaussian case, they recover essentially the √ log n scaling. Finally, in Cai, Jin and Low [8] the authors consider again the estimation of the fraction of significant signal components in the normal means case, and show results beyond consistency, including minimax rates of convergence of the risk. We now proceed with the proof of the theorem and a discussion about tightness of the bounds.…”

Section: Main Results -Detectionmentioning

confidence: 98%

Adaptive sensing performance lower bounds for sparse signal detection and support estimation

Castro¹

2014

Bernoulli

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Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publicationGeneral rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nl providing details and we will investigate your claim. In memory of Yuri IngsterThis paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if x ∈ R n is a sparse vector with s non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds √ 2/s. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds √ 2 log s. Notably there is no dependence on n, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.

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Section: Main Results -Detectionmentioning

confidence: 98%

Adaptive sensing performance lower bounds for sparse signal detection and support estimation

Castro¹

2014

Bernoulli

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“…Inference on the fraction of non-zero components in the model with m = 0 has been studied by Cai et al (2007). They derived minimax rates of convergence for estimators of the fraction and the corresponding confidence sets.…”

Section: (B) Summary Of the Main Resultsmentioning

confidence: 99%

Classification of sparse high-dimensional vectors

Ingster

Pouet

Tsybakov

2009

Phil. Trans. R. Soc. A.

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We study the problem of classification of d-dimensional vectors into two classes (one of which is 'pure noise') based on a training sample of size m. The main specific feature is that the dimension d can be very large. We suppose that the difference between the distribution of the population and that of the noise is only in a shift, which is a sparse vector. For Gaussian noise, fixed sample size m, and dimension d that tends to infinity, we obtain the sharp classification boundary, i.e. the necessary and sufficient conditions for the possibility of successful classification. We propose classifiers attaining this boundary. We also give extensions of the result to the case where the sample size m depends on d and satisfies the condition (log m)/ log d → γ , 0 ≤ γ < 1, and to the case of non-Gaussian noise satisfying the Cramér condition.

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“…The approach of Jin and Cai (2007) also yields a uniformly consistent estimator for the proportion of nonnull effects. In a two-component normal mixture setting, Cai, Jin and Low (2007) proposed an estimator of the proportion and developed a minimax theory for the estimation problem.…”

Section: Estimating the Null Distribution And The Proportion Of The Nmentioning

confidence: 99%