Adaptive estimation over anisotropic functional classes via oracle approach

Lepski, Oleg

doi:10.1214/14-aos1306

Cited by 30 publications

(34 citation statements)

References 70 publications

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“…Interestingly, the set of alternatives that attains the minimax rate is neither sparse nor dense: it presents blocks of signals at different locations. We conjecture that only estimators that incorporate a form of spatial adaptation can be minimax optimal in this regime, as the ones proposed in Lepskii (2015), in delÁlamo et al (2018) and in the present paper.…”

Section: Resultssupporting

confidence: 63%

“…Cavalier (2011)). In contrast, the "slow" regime with rate n − 1 q(d+2β) for q > 1+2/(d+2β) has been observed for the specific case β = 0 in density estimation (Goldenshluger and Lepskii, 2014) and nonparametric regression (Lepskii (2015) and delÁlamo et al (2018)) when estimating over anisotropic Nikolskii classes N s p and Besov classes B s p,t with s < d/p. Moreover, the slow regime explains the recently observed phase transition in the L 2 minimax risk for estimating discretized T V functions in the particular case β = 0, see Sadhanala et al (2016).…”

Section: Resultsmentioning

confidence: 99%

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Total variation multiscale estimators for linear inverse problems

Álamo

Munk

2020

Information and Inference: A Journal of the IMA

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Even though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation (BV ), meaning L 1 functions on a d-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of BV functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper we propose a novel technique for estimating BV functions in an inverse problem setting, and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the L q -risk, for any q ∈ [1, ∞). This is to the best of our knowledge the first convergence result for BV functions in inverse problems in dimension d ≥ 2, and it extends the results of Donoho (1995) in d = 1. Furthermore, our analysis unravels a novel regime for large q in which the minimax rate is slower than n −1/(d+2β+2) , where β is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of BV functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

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Section: Resultssupporting

confidence: 63%

Section: Resultsmentioning

confidence: 99%

Total variation multiscale estimators for linear inverse problems

Álamo

Munk

2020

Information and Inference: A Journal of the IMA

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“…In particular the consideration of compactly supported densities allows to eliminate this zone. The fact that the sparse zone is divided in two sub-domains was discovered in Goldenshluger and Lepski (2014) (bounded case) and in Lepski (2015) (unbounded case). This division can be informally viewed as some "degree of sparsity".…”

Section: Resultsmentioning

confidence: 99%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

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Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

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“…We will use the so-called Lepskii method to derive adaptive estimators. This method introduced in [Lep92] has been successfully applied in many non-parametric estimation problems and we refer to [GL11], [Lep15] for recent contributions using this method. We also refer to [Chi10] for an introduction of this method in different frameworks.…”

Section: Adaptationmentioning

confidence: 99%

Cytometry inference through adaptive atomic deconvolution

Costa

Gadat

Gonnord

et al. 2019

Journal of Nonparametric Statistics

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In this paper we consider a statistical estimation problem known as atomic deconvolution. Introduced in reliability, this model has a direct application when considering biological data produced by flow cytometers. In these experiments, biologists measure the fluorescence emission of treated cells and compare them with their natural emission to study the presence of specific molecules on the cells' surface. They observe a signal which is composed of a noise (the natural fluorescence) plus some additional signal related to the quantity of molecule present on the surface if any. From a statistical point of view, we aim at inferring the percentage of cells expressing the selected molecule and the probability distribution function associated with its fluorescence emission. We propose here an adaptive estimation procedure based on a previous deconvolution procedure introduced by [vEGS08, GvES11]. For both estimating the mixing parameter and the mixing density automatically, we use the Lepskii method based on the optimal choice of a bandwidth using a bias-variance decomposition. We then derive some concentration inequalities for our estimators and obtain the convergence rates, that are shown to be minimax optimal (up to some log terms) in Sobolev classes. Finally, we apply our algorithm on simulated and real biological data.

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Adaptive estimation over anisotropic functional classes via oracle approach

Cited by 30 publications

References 70 publications

Total variation multiscale estimators for linear inverse problems

Total variation multiscale estimators for linear inverse problems

Lower bounds in the convolution structure density model

Cytometry inference through adaptive atomic deconvolution

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