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

arXiv: https://arxiv.org/abs/2502.13486 · HTML: https://mlsanity.com/paper/2502.13486/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 86 references that are in the map.
- First posted on arXiv 19 Feb 2025.

## Builds on (most cited first, 18 of 18)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Nonparametric approximation of conditional expectation operators](https://mlsanity.com/paper/2012.12917/) (2020)
- [The linear conditional expectation in Hilbert space](https://mlsanity.com/paper/2008.12070/) (2020)
- [Robustness to incorrect system models in stochastic control](https://mlsanity.com/paper/1803.06046/) (2018)
- [Near Optimality of Quantized Policies in Stochastic Control Under Weak Continuity Conditions](https://mlsanity.com/paper/1410.6985/) (2014)
- [Continuity of Cost in Borkar Control Topology and Implications on Discrete Space and Time Approximations for Controlled Diffusions under Several Criteria](https://mlsanity.com/paper/2209.14982/) (2022)
- [Continuity of Utility Maximization under Weak Convergence](https://mlsanity.com/paper/1811.01420/) (2018)
- [Near Optimality of Lipschitz and Smooth Policies in Controlled Diffusions](https://mlsanity.com/paper/2405.16223/) (2024)
- [Optimal Control of Partially Observable Piecewise Deterministic Markov Processes](https://mlsanity.com/paper/1706.09142/) (2017)
- [A topology for Team Policies and Existence of Optimal Team Policies in Stochastic Team Theory](https://mlsanity.com/paper/1711.00639/) (2017)
- [Recursive Estimation of Conditional Kernel Mean Embeddings](https://mlsanity.com/paper/2302.05955/) (2023)
- [On Borkar and Young Relaxed Control Topologies and Continuous Dependence of Invariant Measures on Control Policy](https://mlsanity.com/paper/2304.07685/) (2023)
- [Controlled Diffusions under Full, Partial and Decentralized Information: Existence of Optimal Policies and Discrete-Time Approximations](https://mlsanity.com/paper/2311.03254/) (2023)

## Abstract (arXiv)

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.