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 $\mathcal{U}$ 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 $(T^{*})^{-1}(\mathcal{U})$ is of wavelet-type. Based on this, we obtain a multiple test that allows to identify the active components of $\mathcal{U}$, that is, $\langle f,\varphi\rangle\neq0$, $\varphi\in\mathcal{U}$, 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 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, that is, 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.