Characterize contraction in regularized kernel geometries relative to classical contraction notions

Characterize systematically the relationship between contraction in regularized kernel-induced geometries and classical spectral or Wasserstein contraction, and derive criteria for choosing the kernel and regularization parameter adaptively from data.

Background

The paper establishes convergence and learning guarantees under contraction in an MMD or Hilbert-scale geometry, together with an L2(π)L^2(\pi) spectral-gap condition. It also compares these assumptions with total-variation and Wasserstein contraction in selected examples.

A broader unresolved issue is how these different contraction notions relate in general, particularly after regularization changes the kernel-induced geometry. The paper further points to the practical problem of selecting the kernel and regularization level from data, for which no general criteria are provided.

References

Several questions remain open. It would be useful to replace the boundedness assumptions on the feature map and innovations by suitable moment or tail conditions, thereby covering unbounded state spaces and systems with sub-Gaussian or sub-exponential additive noise. Sharper control of the lower-order concentration terms, as well as extensions to misspecified models, vector-valued states, and more general initial distributions, are also of interest. More fundamentally, a systematic understanding of the relation between contraction in regularized kernel-induced geometries and classical spectral or Wasserstein contraction could provide criteria for choosing the kernel and the regularization parameter adaptively from data.

— Markov Chain Convergence in Kernel-Induced Geometry  (2610.13916 - Kachaiev et al., 9 Oct 2026) in Conclusion