Adaptive complexity regularization for linear inverse problems

Adaptive spectral regularizations of high dimensional linear models

This paper focuses on recovering an unknown vector $β$ from the noisy data $Y=Xβ+σξ$, where $X$ is a known $n\times p$-matrix, $ξ$ is a standard white Gaussian noise, and $σ$ is an unknown noise level. In order to estimate $β$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. In this paper, we deal solely with regularization methods based on the so-called ordered smoothers and provide some oracle inequalities in the case, where the noise level is unknown.

非线性统计逆问题中具有过平滑惩罚项的 Tikhonov 正则化

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

针对统计学习中含噪非线性不适定逆问题，本文采用 Hilbert 尺度下的 Tikhonov 正则化重建估计量，并基于算子与先验假设推导收敛速率及 RKHS 误差估计。

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

We study a standard method of regularization by projections of the linear inverse problem $Y=Af+ε$, where $ε$ is a white Gaussian noise, and $A$ is a known compact operator with singular values converging to zero with polynomial decay. The unknown function $f$ is recovered by a projection method using the singular value decomposition of $A$. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particula…

非线性统计逆学习问题的 Tikhonov 正则化收敛性分析

Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

本文针对非线性统计逆学习问题，分析基于再生核 Hilbert 空间的 Tikhonov 正则化估计量，并给出由源条件定义的可容许解类别上的最优一致收敛速率。

论学习逆问题中的最优正则化参数

On Learning the Optimal Regularization Parameter in Inverse Problems

针对逆问题中正则化参数选择难题，本文提出基于经验风险最小化的统计机器学习方法，并证明其在数据充足时可达到尖锐速率且自适应噪声与平滑性。

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。

On the asymptotical regularization for linear inverse problems in presence of white noise

We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation (SDE) system where the inverse of the ending time $T$ naturally plays the role of the squared noise level. The time-continuous framework then allows us to apply classical methods from data assimilation, namely the Kalman-Bucy filter and 3DVAR, and to analyze their behaviour as a regularization method for the original problem. Such treatment offers some connections to the famous asymptotical regularization method, which has not yet been analyzed…

Convex regularization in statistical inverse learning problems

We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$, $n=1,...,N$ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $p$-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov no…

具有随机观测的统计逆学习问题

Statistical inverse learning problems with random observations

概述具有随机实验设计的统计逆问题进展，涵盖线性与非线性，研究不同正则化方案，利用再生核希尔伯特空间结构分析收敛速率，得出极小极大速率，并展示其在PK/PD模型非线性逆问题中的应用。

交叉正则化：通过验证梯度自适应模型复杂度

Cross-regularization: Adaptive Model Complexity through Validation Gradients

本文提出交叉正则化方法，利用验证梯度在训练中自适应调整正则化参数，平衡复杂度与过拟合，并证明其收敛到交叉验证最优，同时支持数据增强和不确定性校准。