针对非独立数据，提出希尔伯特空间中的依赖型伯恩斯坦不等式。
An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications
- 基于希尔伯特空间中的向量过程，构建依赖数据的伯恩斯坦不等式。
- 在协方差算子估计和动态系统算子学习中获得新风险界。
- 适用于平稳与非平稳过程，适合关注高维依赖数据建模的研究者。
从非独立、非同分布数据中学习是统计学习中的长期挑战。本文针对希尔伯特空间中的向量值过程，提出适用于依赖数据的伯恩斯坦不等式。该不等式既可用于平稳过程，也可用于非平稳过程，并利用时间分离变量间相关性快速衰减的特性以提升估计精度。我们通过将这些界应用于希尔伯特-施密特范数下的协方差算子估计及动态系统中的算子学习，获得了新的风险界。最后，通过数值实验展示了这些界在两种场景下的实际意义。
原文摘要 · Abstract (English)
Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.
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