Multiscale scanning in inverse problems

Abstract

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.

Total variation multiscale estimators for linear inverse problems

“…Besides these works, there have been several approaches that implicitly use the WVD idea. We refer to Schmidt-Hieber et al (2013) and Proksch et al (2018) for hypothesis testing in inverse problems, where multiscale dictionaries adapted to the operator T are employed. Another source of inspiration for our work are nonparametric methods that combine variational regularization with multiscale dictionaries.…”

Multiscale Inference and Long-Run Variance Estimation in Non-Parametric Regression with Time Series Errors

“…() and Proksch et al . () have recently used Gaussian approximation results that were derived in Chernozhukov et al . () to analyse multiscale tests for independent data.…”

“…However, it has not been combined with anticoncentration results to approximate the quantiles of the multiscale statistic. As an alternative to strong approximation theory, Eckle et al (2017) and Proksch et al (2018) have recently used Gaussian approximation results that were derived in Chernozhukov et al (2014Chernozhukov et al ( , 2017 to analyse multiscale tests for independent data. Even though it might be possible to adapt these techniques to the case of dependent data, this is not trivial at all as part of the technical arguments and the Gaussian approximation tools strongly rely on the assumption of independence.…”

“…More recently, Proksch et al . () have constructed multiscale tests for inverse regression models. In the context of density estimation, multiscale tests have been investigated in Dümbgen and Walther (), Rufibach and Walther (), Schmidt‐Hieber et al .…”

“…Rank-based multiscale tests for non-parametric regression were proposed in Dümbgen (2002) and Rohde (2008). More recently, Proksch et al (2018) have constructed multiscale tests for inverse regression models. In the context of density estimation, multiscale tests have been investigated in Dümbgen and Walther (2008), Rufibach and Walther (2010), Schmidt-Hieber et al (2013) and Eckle et al (2017) among others.…”