非线性统计逆问题中具有过平滑惩罚项的 Tikhonov 正则化

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。

Hilbert 尺度中的统计逆问题

Statistical Inverse Problems in Hilbert Scales

本文研究 Hilbert 尺度下含一般噪声的非线性统计逆问题，基于条件稳定性与距离函数分析 Tikhonov 正则化，给出直接与重构误差的高概率估计及显式收敛速率。

The cost-free nature of optimally tuning Tikhonov regularizers and other ordered smoothers

We consider the problem of selecting the best estimator among a family of Tikhonov regularized estimators, or, alternatively, to select a linear combination of these regularizers that is as good as the best regularizer in the family. Our theory reveals that if the Tikhonov regularizers share the same penalty matrix with different tuning parameters, a convex procedure based on $Q$-aggregation achieves the mean square error of the best estimator, up to a small error term no larger than $Cσ^2$, where $σ^2$ is the noise level and $C>0$ is an absolute constant. Remarkably, the error term does not d…

A unifying representer theorem for inverse problems and machine learning

The standard approach for dealing with the ill-posedness of the training problem in machine learning and/or the reconstruction of a signal from a limited number of measurements is regularization. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an $\ell_1$-norm (or some variant ther…

Bernstein-von Mises theorems and uncertainty quantification for linear inverse problems

We consider the statistical inverse problem of recovering an unknown function $f$ from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction of $f$ corresponds to a Tikhonov regulariser $\bar f$ with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein-von Mises theorem for a large collection of linear functionals of $f$, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a fre…

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…