Multiscale scanning in inverse problems

TL;DR

This paper introduces a multiscale scanning method for inverse problems that identifies active components in a signal with controlled error rates, applicable to tomography, deconvolution, and imaging, including super-resolution microscopy.

Contribution

It develops a novel multiscale testing procedure with Gaussian approximation and scale penalty, achieving oracle optimality and applicability to various inverse problems.

Findings

Provides uniform confidence statements for coefficients in inverse regression.

Achieves controlled family-wise error rate in component identification.

Demonstrates effectiveness through simulations and super-resolution microscopy application.

Abstract

In this paper we propose a multiscale scanning method to determine active components of a quantity $f$ w.r.t. a dictionary $\mathcal{U}$ from observations $Y$ in an inverse regression model $Y=Tf+\xi$ with linear operator $T$ and general random error $\xi$. To this end, we provide uniform confidence statements for the coefficients $\langle \varphi, f\rangle$, $\varphi \in \mathcal U$, under the assumption that $(T^*)^{-1} \left(\mathcal U\right)$ is of wavelet-type. Based on this we obtain a multiple test that allows to identify the active components of $\mathcal{U}$, i.e. $\left\langle f, \varphi\right\rangle \neq 0$, $\varphi \in \mathcal U$, 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 $T = \text{id}$ 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.