SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3504-3529, January 2020.
In this paper we study the relation between convergence rates of spectral regularization methods under Hölder-type source conditions resulting from the theory of ill-posed inverse problems, when the noise level $\delta$ goes to 0, and convergence rates resulting from statistical kernel learning, when the number of samples n goes to infinity. Toward this aim, we introduce a family of hybrid estimators in the statistical learning context whose convergence rates have the following properties: first, they are equal to those of spectral methods, and second, they are connected to the rates of spectral regularization in ill-posed inverse problems, provided that a suitable inverse proportionality relation between n and $\delta$ holds true. This family of estimators allows us to convert upper rates depending on $n$ to upper rates depending on $\delta$ and to convert lower rates vice versa, quantifying their deviation. The analysis is carried out under general source conditions in the case the rank of the forward operator is both finite and infinite, and, in the latter case, both by not making any assumptions on the eigenvalues and by assuming a polynomial eigenvalue decay.

中文翻译：

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谱正则化方法的收敛速度：不适定逆问题与统计核学习的比较

SIAM数值分析杂志，第58卷，第6期，第3504-3529页，2020年1月。
在本文中，我们研究了由不适定逆问题理论（当噪声水平$ \ delta $变为0时）产生的Hölder源条件下的光谱正则化方法的收敛速度与统计核所导致的收敛速度之间的关系。学习时，样本数n变为无穷大。为了实现这一目标，我们在统计学习环境中引入了一系列混合估计量，其收敛速度具有以下特性：首先，它们等于频谱方法的收敛速度；其次，它们与不适定条件下的频谱正则化速率相关。假设n和$ \ delta $之间的合适的反比例关系成立，则提出反问题。这个估算器族使我们能够将取决于$ n $的较高利率转换为取决于$ \ delta $的较高利率，反之亦然，将较低利率转换为量化它们的偏差。在前向算子的秩为有限和无限的情况下，在一般源条件下进行分析，在后一种情况下，通过不对特征值进行任何假设并通过多项式特征值衰减来进行分析。