提出新型核均值嵌入拓扑，统一处理随机核的弱与强形式，助力模型学习与控制优化。
Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning
- 基于希尔伯特空间构造随机核的弱/强拓扑，支持灵活建模
- 弱形式下与广义策略空间兼容，具相对紧性但不闭合
- 强形式适用于鲁棒控制与学习理论，适合优化与逼近研究
本文引入一种新的拓扑——核均值嵌入拓扑，用于描述随机核，包含弱与强两种形式。该拓扑定义在从信号空间到概率测度空间的博赫纳可积函数空间上，具备希尔伯特空间结构。弱形式下，其在松弛策略空间中具有实用性，与杨窄拓扑和博卡尔（或 $ w^* $）拓扑存在关联，并确立了等价性。研究表明，尽管 $ w^* $ 拓扑与核均值嵌入拓扑均具相对紧性，但均不闭合；而杨窄拓扑虽闭合，却缺乏相对紧性。强形式则为具有显式鲁棒性的随机核模型空间提供了合适的拓扑，对折扣或平均成本准则下的最优随机控制具有学习理论意义。因此，该拓扑兼具优化与近似（弱形式）及鲁棒性（强形式）分析优势，适用于多种应用。
原文摘要 · Abstract (English)
We introduce a novel topology, called Kernel Mean Embedding Topology, for stochastic kernels, in a weak and strong form. This topology, defined on the spaces of Bochner integrable functions from a signal space to a space of probability measures endowed with a Hilbert space structure, allows for a versatile formulation. This construction allows one to obtain both a strong and weak formulation. (i) For its weak formulation, we highlight the utility on relaxed policy spaces, and investigate connections with the Young narrow topology and Borkar (or \( w^* \))-topology, and establish equivalence properties. We report that, while both the \( w^* \)-topology and kernel mean embedding topology are relatively compact, they are not closed. Conversely, while the Young narrow topology is closed, it lacks relative compactness. (ii) We show that the strong form provides an appropriate formulation for placing topologies on spaces of models characterized by stochastic kernels with explicit robustness and learning theoretic implications on optimal stochastic control under discounted or average cost criteria. (iii) We thus show that this topology possesses several properties making it ideal to study optimality and approximations (under the weak formulation) and robustness (under the strong formulation) for many applications.
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