Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning

2025-02-19

AI Summary

Random kernels lack topological adaptability to structured signal spaces. Method: We propose kernel mean embedding topologies—weak and strong variants—defined on the Bochner integrable function space mapping signals to probability measures in a reproducing kernel Hilbert space (RKHS). This work pioneers the integration of kernel mean embeddings with random kernel analysis, yielding a novel topology that simultaneously preserves Hilbert structure and statistical computability. Theoretically, the weak topology is shown equivalent to the Young narrow topology and Borkar’s w*-topology, while the strong topology ensures compactness and closedness. Contributions: (i) Provides convergence guarantees for robust policy learning in discounted and average-cost optimal stochastic control; (ii) Unifies compactness, closedness, and robustness characterizations of policy spaces; (iii) Enables RKHS-based approximation from simulation data; (iv) Establishes a unified topological framework for policy optimization, model continuity, and stability analysis.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 show that this topology possesses several properties making it ideal to study optimality, approximations, robustness and continuity properties. In particular, the kernel mean embedding topology has a Hilbert space structure, which is particularly useful for approximating stochastic kernels through simulation data.

Problem

• Introduces Kernel Mean Embedding Topology for stochastic kernels.
• Explores weak and strong formulations in model learning.
• Examines robustness and continuity in optimal stochastic control.

Innovation

• Introduces Kernel Mean Embedding Topology
• Weak and strong formulations for kernels
• Hilbert space structure for approximations