On additive averaging kernels for finite Markov chains
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arXiv
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| Format: | Preprint |
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2026
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| author | Lim, Ryan J. Y. Choi, Michael C. H. |
| author_facet | Lim, Ryan J. Y. Choi, Michael C. H. |
| contents | We study additive mixtures of Markov kernels of the form $A_α= αP + (1-α)G$, where $α\in [0,1]$, $P$ is a baseline sampler and $G$ is a Gibbs kernel induced by a partition of the state space. We first motivate the study of $A_α$, 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 formulas 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 $A_α$ 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 $α$ can significantly accelerate convergence in total variation distance. We observe a consistent trade-off between local exploration and global averaging, with intermediate values of $α$ achieving the best performance across regimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12334 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On additive averaging kernels for finite Markov chains Lim, Ryan J. Y. Choi, Michael C. H. Probability Information Theory Combinatorics Optimization and Control Computation 60J10, 60J22, 65C40, 90C27, 94A15, 94A17 We study additive mixtures of Markov kernels of the form $A_α= αP + (1-α)G$, where $α\in [0,1]$, $P$ is a baseline sampler and $G$ is a Gibbs kernel induced by a partition of the state space. We first motivate the study of $A_α$, 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 formulas 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 $A_α$ 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 $α$ can significantly accelerate convergence in total variation distance. We observe a consistent trade-off between local exploration and global averaging, with intermediate values of $α$ achieving the best performance across regimes. |
| title | On additive averaging kernels for finite Markov chains |
| topic | Probability Information Theory Combinatorics Optimization and Control Computation 60J10, 60J22, 65C40, 90C27, 94A15, 94A17 |
| url | https://arxiv.org/abs/2604.12334 |