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.

翻译：我们研究形如$A_α= αP + (1-α)G$的马尔可夫核的加性混合，其中$α\in [0,1]$，$P$是基础采样器，$G$是由状态空间划分诱导的吉布斯核。首先，我们阐述研究$A_α$的动机，该核可解释为 lifted 马尔可夫链的投影。随后，我们考虑在两种目标下最小化趋于平稳的距离：平方弗罗贝尼乌斯范数和库尔贝克-莱布勒散度。对于弗罗贝尼乌斯目标，我们推导出显式的迹公式，并识别出一类刻画最优两块划分的 Cheeger 型泛函。这导出一个可分解为子模差结构的组合优化问题，从而可通过最大最小化算法高效逼近。我们还得到了由$P$的绝对谱隙控制的几何衰减率。对于库尔贝克-莱布勒散度，我们建立了基于凸性的界，表明$A_α$的散度受$P$和$G$两者散度的控制，从而将划分选择约简为吉布斯分量。在居里-外斯模型上的数值实验表明，适当选择划分和参数$α$可显著加速全变差距离下的收敛。我们观察到局部探索与全局平均之间存在一致的权衡，其中中等$α$值在各状态下实现最佳性能。