Risk hull method and regularization by projections of ill-posed inverse problems

Regularization of linear ill-posed problems involving multiplication operators

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-…

线性不适定积分方程的自适应最优正则化

Adaptive optimal regularization of the linear ill posed integral equations

针对第一类线性积分方程，提出一种自适应渐近最优（在L2范数意义下阶最优）的解估计方法，并构建自适应置信区域。

在病态问题中选择正则化参数的风险估计器——性质与局限性

Risk Estimators for Choosing Regularization Parameters in Ill-Posed Problems - Properties and Limitations

本文研究了用于病态问题中选择正则化参数的风险估计器的性质与局限性，分析了SURE和GSURE两种方法在有限维线性Tikhonov正则化中的表现，并探讨了其在高维问题下的渐进行为。

Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation

In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelity term and a regularization term formed of the L1-norm of the correlations between the sought after signal and atoms in a given (generally overcomplete) dictionary. The L1-sparsity analysis prior is …