On additive averaging kernels for finite Markov chains

Abstract

We study additive mixtures of Markov kernels of the form $A_{\hat{I}}\pm =\hat{I}\pm P+(1-\hat{I}\pm )G$, where $\hat{I}\pm \in [0,1]$, $P$ is a baseline sampler and $G$ is a Gibbs kernel induced by a partition of the state space. We first motivate the study of $A_{\hat{I}}\pm$, which can be interpreted as the projection of a lifted Markov chain. We then consider the minimisation of distance to stationarity under two objectives: the squared Frobenius norm and the Kullback-Leibler (KL) divergence. For the Frobenius objective, we derive explicit trace formulae and identify a Cheeger-type functional that characterises optimal two-block partitions. This yields a structured combinatorial optimisation problem admitting a difference-of-submodular decomposition, enabling efficient approximation via majorisation-minimisation. We also obtain geometric decay rates governed by the absolute spectral gap of $P$. For the KL divergence, we establish convexity-based bounds showing that the divergence of $A_{\hat{I}}\pm$ is controlled by those of both $P$ and $G$, thereby reducing partition selection to the Gibbs component. Numerical experiments on the Curie-Weiss model demonstrate that suitable choice of both the partition and the parameter $\hat{I}\pm$ can significantly accelerate convergence in total variation distance. We observe a consistent trade-off between local exploration and global averaging, with intermediate values of $\hat{I}\pm$ achieving the best performance across regimes.

32 pages, 5 figures

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