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Multiscale scanning in inverse problems. (English) Zbl 1410.62064

This paper introduces a novel method to detect active components of an unknown function of interest w.r.t. a prescribed dictionary \(\left\{\varphi_i\right\}_{i\in I}\) from indirect observations \(Y_j = T f\left(x_j\right) + \xi_j\). Here, \(T\) is a bounded linear operator acting between proper Hilbert spaces, \(x_j\) are deterministic sampling points, and \(\xi_j\) are independent errors. The method is based on a multiscale test statistic, which allows to test \(\left\langle \varphi_i, f\right\rangle = 0\) vs. \(\left|\left\langle \varphi_i, f\right\rangle\right|>0\) simultaneously over all subsets \(J \subset I\). The authors present a unified asymptotic theory for the considered global test statistic, which allows to calibrate the corresponding multiple hypothesis test universally and independent from the specific data set and reveals asymptotic minimax optimality. The method is then applied to an inverse problem from super-resolution fluorescence microscopy and its finite sample performance is investigated in a simulation study.

MSC:

62G10 Nonparametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62G32 Statistics of extreme values; tail inference
62P10 Applications of statistics to biology and medical sciences; meta analysis

Software:

essHist
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References:

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