发表机构
Technical University of Denmark; The Chinese University of Hong Kong, Shenzhen（丹麦技术大学; 香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究LTI SDE诱导核的SSS结构，提出线性复杂度递归算法用于GP回归及系统辨识中的核估计，涵盖分解、求解、梯度与采样，并经数值实验验证。
AI 中文摘要
我们研究由线性时不变（LTI）随机微分方程（SDE）诱导的核函数，这种方法将关于待估计函数的先验知识直接嵌入SDE的状态空间描述中。这种核设计视角在许多核函数发挥核心作用的场景中具有相关性，包括统计学中的核方法、机器学习中的高斯过程（GP）回归以及系统辨识中基于核的正则化方法。由此产生的SDE诱导核函数类别包含若干具有实际意义的核，例如具有半整数光滑性的Matérn核、样条核以及系统辨识中使用的相关核。与现有的针对这些核的线性复杂度GP回归方法（通常基于潜在状态上的卡尔曼滤波和Rauch–Tung–Striebel平滑）相比，我们直接处理观测协方差矩阵的SSS表示。这为GP回归所需的矩阵运算提供了线性复杂度的递归算法，包括分解、线性求解、对数行列式、后验协方差、对数边际似然梯度以及后验采样；相同的生成器级递归只需稍作修改即可扩展到系统辨识中基于核的正则化方法的类似计算。与先前在GP回归中使用SSS表示的工作相比，我们在三个应用相关的情形中推导了解析边际似然梯度的高效递归算法，并开发了用于后验计算和Matheron型采样的新递归。数值实验验证了所提递归的准确性，并展示了代表性SDE诱导核的后验推断和采样。
英文摘要
We study kernel functions induced by a linear time-invariant (LTI) stochastic differential equation (SDE), an approach that embeds prior knowledge about the function to be estimated directly into the state-space description of the SDE. This kernel-design perspective is relevant across the many settings where kernels play a central role, including kernel methods in statistics, Gaussian process (GP) regression in machine learning, and kernel-based regularization methods in system identification. The resulting class of SDE-induced kernels includes several kernels of practical interest, such as Matérn kernels with half-integer smoothness, spline kernels, and related kernels used in system identification. In contrast to existing linear-time GP regression methods for these kernels, which are typically based on Kalman filtering and Rauch--Tung--Striebel smoothing on the latent state, we work directly with the SSS representation of the observed covariance matrix. This yields linear-complexity recursive algorithms for the matrix operations required in GP regression, including factorization, linear solves, log determinants, posterior covariance, log-marginal-likelihood gradients, and posterior sampling; the same generator-level recursions extend, with only minor modifications, to the analogous computations in kernel-based regularization methods for system identification. Compared with previous uses of SSS representations in GP regression, we derive efficient recursive algorithms for analytic marginal-likelihood gradients in three application-relevant cases, and we develop new recursions for posterior computations and Matheron-type sampling. Numerical experiments verify the accuracy of the proposed recursions and illustrate posterior inference and sampling for representative SDE-induced kernels.
