用核方法直接学习随机微分方程的漂移与扩散项，无需似然计算。
The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations
- 基于向量与算子型占有核，分两步估计SDE的漂移与扩散函数。
- 在模拟数据和阿尔茨海默病脑成像数据上均实现高精度预测。
- 避免复杂似然计算，适合轨迹数据建模与生物医学应用。
我们提出一种新型核方法，用于学习多变量随机微分方程（SDE）。该方法分为两步：先估计漂移项函数，再基于已知漂移估计（矩阵值）扩散函数。占有核是再生核希尔伯特空间（RKHS）上的积分泛函，能聚合轨迹信息。本方法利用向量值占有核估计过程的漂移成分；对于扩散估计，引入算子值占有核，将辅助矩阵值函数作为半正定算子进行估计，从而简便导出扩散项。该方法避免了传统SDE学习中难以处理的似然问题，通过最小化重构误差目标函数进行优化。我们设计了一种简单高效的学习流程，并借助Fenchel对偶性提升计算效率。在模拟基准数据集和健康及阿尔茨海默病患者的淀粉样蛋白成像真实数据上进行了验证。
原文摘要 · Abstract (English)
We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix-valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate. This enables us to avoid common challenges in SDE learning, such as intractable likelihoods, by optimizing a reconstruction-error-based objective. We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency. We validate the method on simulated benchmarks and a real-world dataset of Amyloid imaging in healthy and Alzheimer's disease subjects.
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