逆问题中的多尺度扫描
Multiscale scanning in inverse problems
AI总结：
本文提出多尺度扫描方法，用于确定逆回归模型中量f相对于字典U的活跃成分。通过提供统一的置信陈述，实现对活跃成分的识别，同时控制错误率。方法基于Gaussian近似和自适应的尺度惩罚，适用于断定问题，并在医学成像中展示应用。
摘要：
在本文中，我们提出了一种多尺度扫描方法，用于从观测Y中确定逆回归模型Y=Tf+ξ中量f相对于字典U的活跃成分。为此，我们在假设(T*)^{-1}(U)为小波型的情况下，为系数⟨φ,f⟩，φ∈U提供统一的置信陈述。基于此，我们获得一个多重检验，允许在受控的家族错误率下识别U中的活跃成分，即⟨f,φ⟩≠0，φ∈U。我们的结果依赖于对底层多尺度统计量的高斯近似，结合一种自适应的尺度惩罚，以适应问题的病态性。尺度惩罚还确保统计量的分布弱收敛到Gumbel极限。我们详细讨论了断定和反卷积的重要特殊情况。此外，当T=id且字典由不同大小的移动窗口（尺度）组成时的回归情况也包含在内，扩展了此前在该设置下的结果。我们证明了我们的方法满足Oracle最优性，即在正确尺度下，它达到与单尺度检验程序相同的渐近功效。模拟支持我们的理论，我们展示了该方法作为成像工具的潜力。作为特定应用，我们讨论了超分辨率显微镜，并分析了实验STED数据以定位单个DNA折纸结构。
英文摘要：
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 $Y=Tf+ξ$ with linear operator $T$ and general random error $ξ$. To this end, we provide uniform confidence statements for the coefficients $\langle φ, f\rangle$, $φ\in \mathcal U$, 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. $\left\langle f, φ\right\rangle \neq 0$, $φ\in \mathcal U$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
