Multiscale inference for multivariate deconvolution

Eckle, Konstantin; Bissantz, Nicolai; Dette, Holger

doi:10.48550/arxiv.1611.05201

Cited by 3 publications

(4 citation statements)

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“…Note that the calibration proposed by Dümbgen and Spokoiny (2001) for direct regression problems (which is frequently employed in multiscale procedures, see e.g. Rohde (2008); Walther (2010); Schmidt-Hieber et al ( 2013); Eckle et al (2016)) is tailored to a continuous observation setting in which all scales within a range p0, as, a P R `are considered. If this calibration is used in a discrete setting like (1), the overall test-statistic converges to a degenerate limit, since the largest scale h max has to satisfy h max Ñ 0 as n Ñ 8, otherwise the finite sample approximations do not converge to their continuous counterparts.…”

Section: Multiscale Inverse Scanning Test: Miscatmentioning

confidence: 99%

“…Instead, it becomes necessary to employ probe functionals ϕ i " ϕ i,n (again dependent on the discretization level n, but this dependence will be suppressed whenever not relevant below), which are compatible with the operator T and hence allow for transportation of "local" information from T f back to xf, ϕ i y. If the probe functionals ϕ i are chosen properly, the values xf, ϕ i y hold information about "local" features of f , e. g. in form of a wavelet-type analysis, see also Schmidt-Hieber et al (2013); Eckle et al (2016), who infer on shape characteristics in i.i.d. density deconvolution.…”

Section: Introductionmentioning

confidence: 99%

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Multiscale scanning in inverse problems

2018

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In this paper we propose a multiscale scanning method to determine active components of a quantity f w.r.t. a dictionary U from observations Y in an inverse regression model Y " T f`ξ with linear operator T and general random error ξ. To this end, we provide uniform confidence statements for the coefficients xϕ, f y, ϕ P U, under the assumption that pT˚q´1 pUq is of wavelet-type. Based on this we obtain a multiple test that allows to identify the active components of U, i.e. xf, ϕy ‰ 0, ϕ P U, at controlled, family-wise error rate. Our results rely on a Gaussian approximation of the underlying multiscale statistic with a novel scale penalty adapted to the ill-posedness of the problem. The scale penalty furthermore ensures weak convergence of the statistic's distribution towards a Gumbel limit under reasonable assumptions. The important special cases of tomography and deconvolution are discussed in detail. Further, the regression case, when T " id and the dictionary consists of moving windows of various sizes (scales), is included, generalizing previous results for this setting. We show that our method obeys an oracle optimality, i.e. it attains the same asymptotic power as a single-scale testing procedure at the correct scale. Simulations support our theory and we illustrate the potential of the method as an inferential tool for imaging. As a particular application we discuss super-resolution microscopy and analyze experimental STED data to locate single DNA origami.

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Section: Multiscale Inverse Scanning Test: Miscatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiscale scanning in inverse problems

2018

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“…data, the distribution of X is recovered by filtering the received observations to compensate for the convolution using Fourier inversion and Kernel based methods, see [Devroye, 1989, Liu and Taylor, 1989, Stefanski and Carroll, 1990 for some early nonparametric deconvolution methods and [Carroll andHall, 1988, Fan, 1991] for minimax rates. On the other hand, more recent works were dedicated to multivariate deconvolution problems such as [Comte and Lacour, 2013] for kernel density estimators, [Sarkar et al, 2018] for a Bayesian approach or [Eckle et al, 2016] for a multiscale based inference. In all these works, deconvolution is solved under two restrictive assumptions: (a) the distribution of the noise is assumed to be known and (b) this distribution is assumed to be such that its Fourier transform is nowhere vanishing.…”

Section: Introductionmentioning

confidence: 99%

Deconvolution with unknown noise distribution is possible for multivariate signals

Gassiat,

Corff,

Lehéricy

2020

Preprint

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This paper considers the deconvolution problem in the case where the target signal is multidimensional and no information is known about the noise distribution. More precisely, no assumption is made on the noise distribution and no samples are available to estimate it: the deconvolution problem is solved based only on the corrupted signal observations. We establish the identifiability of the model up to translation when the signal has a Laplace transform with an exponential growth smaller than 2 and when it can be decomposed into two dependent components. Then, we propose an estimator of the probability density function of the signal without any assumption on the noise distribution. As this estimator depends of the lightness of the tail of the signal distribution which is usually unknown, a model selection procedure is proposed to obtain an adaptive estimator in this parameter with the same rate of convergence as the estimator with a known tail parameter. Finally, we establish a lower bound on the minimax rate of convergence that matches the upper bound.

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“…Hence, our study contributes to the deconvolution theory by treating the multivariate case; in particular, our techniques for the lower bounds might be of interest. To our knowledge, only Masry [39], Eckle et al [20], and Lepski and Willer [35,36] have studied the setting of multivariate deconvolution. They deal with a different problem, namely that of nonparametric estimation of the density of X j or its geometric features when the distribution of ε j is known.…”

Section: Introductionmentioning

confidence: 99%

Sparse covariance matrix estimation in high-dimensional deconvolution

Belomestny¹,

Trabs²,

Tsybakov³

2019

Bernoulli

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We study the estimation of the covariance matrix Σ of a p-dimensional normal random vector based on n independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of Σ. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in n/ log p. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.MSC 2010 subject classifications: Primary 62H12; secondary 62F12, 62G05.

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Multiscale inference for multivariate deconvolution

Cited by 3 publications

References 0 publications

Multiscale scanning in inverse problems

Multiscale scanning in inverse problems

Deconvolution with unknown noise distribution is possible for multivariate signals

Sparse covariance matrix estimation in high-dimensional deconvolution

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