算子中核贝叶斯学习的数据自适应先验
A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators
AI总结：
针对算子中核学习的病态反问题，本文提出一种基于L曲线自适应选择超参数的数据自适应先验，克服固定先验在小噪声下发散的问题，并在Toeplitz矩阵和积分算子上验证其稳定性。
摘要：
核在表示非局部依赖性方面十分高效，并被广泛用于设计函数空间之间的算子。因此，从数据中学习算子中的核是一个具有普遍意义的反问题。由于非局部依赖性的存在，该反问题可能严重不适定，并具有依赖于数据的奇异求逆算子。贝叶斯方法通过非退化先验克服不适定性。然而，当观测噪声变小时，如果数据在求逆算子零特征值的特征空间中引起扰动，固定的非退化先验会导致后验均值发散。我们引入一种数据自适应先验，以获得稳定的后验，其后验均值始终具有小噪声极限。该数据自适应先验的协方差是求逆算子，并带有一个通过L曲线方法根据数据自适应选择的超参数。此外，我们对数据自适应先验的计算实践进行了详细分析，并在Toeplitz矩阵和积分算子上进行了演示。数值测试表明，在存在四种误差类型中的任何一种时——离散化误差、模型误差、部分观测和错误的噪声假设——固定先验都可能导致后验均值发散。相比之下，数据自适应先验始终能获得具有小噪声极限的后验均值。
英文摘要：
Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem can be severely ill-posed with a data-dependent singular inversion operator. The Bayesian approach overcomes the ill-posedness through a non-degenerate prior. However, a fixed non-degenerate prior leads to a divergent posterior mean when the observation noise becomes small, if the data induces a perturbation in the eigenspace of zero eigenvalues of the inversion operator. We introduce a data-adaptive prior to achieve a stable posterior whose mean always has a small noise limit. The data-adaptive prior's covariance is the inversion operator with a hyper-parameter selected adaptive to data by the L-curve method. Furthermore, we provide a detailed analysis on the computational practice of the data-adaptive prior, and demonstrate it on Toeplitz matrices and integral operators. Numerical tests show that a fixed prior can lead to a divergent posterior mean in the presence of any of the four types of errors: discretization error, model error, partial observation and wrong noise assumption. In contrast, the data-adaptive prior always attains posterior means with small noise limits.
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