广义样条与高斯过程
Generalized Splines and Gaussian Processes
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中文总结 AI 辅助
该研究将有限维线性逆问题中最小均方误差估计与正则化最小二乘的等价性扩展至无限维场景，提出含白化/正则化算子的广义形式体系，可恢复已知等价实例并覆盖通用贝叶斯方法。
中文摘要 AI 辅助
对于变量为高斯分布的有限维线性逆问题，众所周知，最小均方误差估计量采用正则化最小二乘数据拟合的形式。在本章中，我们证明该等价性可扩展至更广泛的无限维场景，其中广义样条充当线性回归量，核空间S上的广义高斯过程对应高斯随机向量。该扩展的范围与从经典函数概念到分布（亦称“广义函数”）概念的转换性质相同。我们的形式体系包含白化/正则化算子L:S→S'，其连续扩张诱导出原生希尔伯特空间H⊂S'，该空间在我们的表征中起核心作用。论述在很大程度上是自包含的，且具有极强的通用性和强大性。它允许恢复此类等价关系的所有已知实例；特别是Kailath及其学生开发的涉及新息与再生核希尔伯特空间的方法，以及分数阶样条与Mandelbrot分数布朗运动（分形）之间的数学对应关系，其中前者是后者的最优估计量。它还涵盖了用于求解无限维逆问题的通用贝叶斯方法。
英文摘要
For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.
