用核方法学习非线性系统的动态距离，实现轨迹分类。
Minimum distance classification for nonlinear dynamical systems
- 基于核函数构建特征空间，通过柯尔普曼算子线性化动态系统。
- 在特征空间中计算轨迹间距离，支持高维甚至无限维数据。
- 可融合先验动力学知识，适用于混沌、手写轨迹等场景。
针对由不同非线性动力系统生成的轨迹数据分类问题，我们提出 Dynafit：一种基于核方法的学习框架，用于学习训练轨迹与底层动力系统之间的距离度量。新观测样本根据该度量被分配到最相似的动力系统类别。学习算法通过近似柯尔普曼算子，在与核函数相关的（可能无限）特征空间中全局线性化动力系统。利用机器学习中的核技巧，距离度量可在特征空间中独立于维度计算。此外，当存在部分动力学先验知识时，核函数可被定制以融入这些信息。Dynafit 可应用于多种涉及非线性动力系统和传感器的分类任务。我们在三个示例中验证其有效性：基于逻辑映射的混沌检测、手写动态识别以及视觉动态纹理识别。
原文摘要 · Abstract (English)
We address the problem of classifying trajectory data generated by some nonlinear dynamics, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method for learning a distance metric between training trajectories and the underlying dynamics. New observations are assigned to the class with the most similar dynamics according to the learned metric. The learning algorithm approximates the Koopman operator which globally linearizes the dynamics in a (potentially infinite) feature space associated with a kernel function. The distance metric is computed in feature space independently of its dimensionality by using the kernel trick common in machine learning. We also show that the kernel function can be tailored to incorporate partial knowledge of the dynamics when available. Dynafit is applicable to various classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness on three examples: chaos detection with the logistic map, recognition of handwritten dynamics and of visual dynamic textures.
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