标题： 自动再生核和正则化用于学习卷积核 显示英文标题

标题： Automatic reproducing kernel and regularization for learning convolution kernels

摘要： 从数据中学习算子的卷积核出现在许多应用中，并代表了一个广受关注的不适定逆问题。 在先验信息很少的情况下，核方法提供了一种自然的非参数化方法并带有正则化。 然而，主要挑战在于选择适当的再生核，特别是当算子和数据变化时。 我们表明，输入数据和卷积算子本身会诱导一个自动的数据自适应RKHS（DA-RKHS），从而避免了手动选择核。 特别是，当观测数据是离散且有限时，存在一个有限的自动基函数集合，足以在DA-RKHS中表示估计量，包括最小范数最小二乘、Tikhonov和共轭梯度估计量。 我们利用自动基函数开发了Tikhonov以及可扩展的迭代和混合算法。 在积分、非局部和聚合算子上的数值实验表明，我们的自动RKHS正则化方法始终优于具有预选核的标准岭回归和高斯过程方法。 显示英文摘要

摘要： Learning convolution kernels in operators from data arises in numerous applications and represents an ill-posed inverse problem of broad interest. With scant prior information, kernel methods offer a natural nonparametric approach with regularization. However, a major challenge is to select a proper reproducing kernel, especially as operators and data vary. We show that the input data and convolution operator themselves induce an automatic, data-adaptive RKHS (DA-RKHS), obviating manual kernel selection. In particular, when the observation data is discrete and finite, there is a finite set of automatic basis functions sufficient to represent the estimators in the DA-RKHS, including the minimal-norm least-squares, Tikhonov, and conjugate-gradient estimators. We develop both Tikhonov and scalable iterative and hybrid algorithms using the automatic basis functions. Numerical experiments on integral, nonlocal, and aggregation operators confirm that our automatic RKHS regularization consistently outperforms standard ridge regression and Gaussian process methods with preselected kernels.