Title: A Doeblin-Anchored Contrastive Chart for Learning Markov Transition Kernels

Original: arXiv:2606.02232v1 Announce Type: new Abstract: Learning a Markov transition model is not merely conditional density estimation: the learned object must be a valid transition kernel before it is iterated in downstream dynamics. This paper introduces a Doeblin-anchored contrastive chart, a statistical-to-dynamical coordinate framework for learning transition kernels from contrastive objectives. Given a restart law and an anchor strength, the chart mixes the target transition with the restart law. The resulting anchored kernel is simultaneously a Doeblin-minorized Markov kernel, the positive conditional law in a binary contrastive experiment, and an explicitly invertible coordinate for the original transition law. We prove that the anchored contrastive risk identifies the anchored transition density and calibrates excess risk to density error. Since inversion of a learned score may produce a signed or unnormalized object, we introduce a measurable Markovization operator that restores kernel validity while preserving integrated $L^1$ accuracy up to a constant factor. Oracle inequalities and H\"older--ReLU approximation bounds yield nonparametric rates for independent transition pairs. For stationary geometrically $\beta$-mixing trajectories, a conservative thinning-and-coupling extension yields the same reconstruction interface with an effective sample size. Occupancy-weighted perturbation bounds transfer one-step kernel error to finite-horizon marginal, path-law, and occupation-measure errors under explicit coverage.

Rewrite: Title: Learning Markov Transition Kernels via a Doeblin-anchored Contrastive Chart

Abstract: Acquiring a Markov transition model involves more than standard conditional density estimation; the resulting model must constitute a legitimate transition kernel prior to its application in subsequent dynamical systems. To address this, we present a Doeblin-anchored contrastive chart, which serves as a coordinate framework bridging statistical and dynamical perspectives for deriving transition kernels through contrastive learning. By combining the target transition with a specified restart law—modulated by an anchor strength—the chart generates an anchored kernel. This anchored kernel functions concurrently as a Doeblin-minorized Markov kernel, the positive conditional distribution within a binary contrastive setup, and an explicitly invertible representation of the original transition law. We demonstrate that the anchored contrastive risk successfully recovers the anchored transition density, with excess risk directly linked to density error. Because inverting a learned score can result in a signed or unnormalized entity, we propose a measurable Markovization operator. This operator reinstates the validity of the kernel while maintaining integrated $L^1$ accuracy within a constant factor. Utilizing oracle inequalities and H\"older--ReLU approximation bounds, we establish nonparametric convergence rates for independent transition pairs. In the context of stationary geometrically $\beta$-mixing trajectories, a conservative extension based on thinning and coupling provides an equivalent reconstruction interface, adjusted for effective sample size. Furthermore, occupancy-weighted perturbation bounds allow for the propagation of one-step kernel errors to finite-horizon marginal, path-law, and occupation-measure errors, contingent upon explicit coverage conditions.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC