发表机构
Automatic Control Laboratory, EPFL（洛桑联邦理工学院自动控制实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对向量值核岭回归，在输出受有界噪声干扰且算子属于再生核希尔伯特空间的假设下，推导了估计量与真实值间逐点确定性误差界，并通过数值模拟验证。研究填补了非线性向量值算子学习理论空白，对系统辨识有重要意义。
AI 中文摘要
虽然针对标量值函数的基于核的学习方法已被广泛研究，但针对非线性向量值算子的工作相对较少。这类具有向量值输出（例如状态向量和轨迹）的算子常见于系统辨识问题中。在本文中，我们研究在向量值核岭回归框架下学习向量值算子的问题。我们假设输出取值于任意希尔伯特空间，并受到范数有界的加性噪声的干扰。在未知算子属于向量值再生核希尔伯特空间的假设下，我们推导了核岭回归估计量与真实值之间在相应希尔伯特空间范数下的逐点确定性界。我们展示了两种不同场景下的数值模拟，以验证理论结果。
英文摘要
While kernel-based learning methods for scalar-valued functions have been extensively studied, comparatively little work considers nonlinear vector-valued operators. Such operators with vector-valued outputs, e.g., state vectors and trajectories, commonly arise in system identification problems. In this paper, we study the problem of learning a vector-valued operator under the framework of vector-valued kernel ridge regression. We assume that the outputs take values in an arbitrary Hilbert space and are corrupted by additive noise with bounded norm. Under the assumption that the unknown operator belongs to a vector-valued reproducing kernel Hilbert space, we derive a pointwise deterministic bound on the discrepancy between the kernel ridge regression estimator and the ground truth in the corresponding Hilbert space norm. Numerical simulations under two different scenarios are presented to validate the theoretical results.
