We realize the half-steps of a general class of Markov chains as alternating projections with respect to the reverse Kullback-Leibler divergence between sets of joint probability distributions which admit an information geometrical property known as autoparallelism. We exhibit both a weak and strong duality between the Markov chains defined by the even and odd half-steps of the alternating projection scheme, which manifests as an equivalence of entropy decay statement regarding the chains, with a precise characterization of the entropy decrease at each half-step. We apply this duality to several Markov chains of interest, and obtain either new results or short, alternative proofs for mixing bounds on the dual chain, with the Swendsen-Wang dynamics serving as a key example. Additionally, we draw parallels between the half-steps of the relevant Markov chains and the Sinkhorn algorithm from the field of entropically regularized optimal transport, which are unified by the perspective of alternating projections with respect to an \alpha-divergence on the probability simplex.

翻译：我们将一类广义马尔可夫链的半步实现为关于反向Kullback-Leibler散度的交替投影，该投影作用于具有信息几何中称为自平行性性质的联合概率分布集合之间。我们展示了由交替投影方案中偶数半步与奇数半步所定义的马尔可夫链之间的弱对偶与强对偶关系，这种关系表现为关于两条链的熵衰减陈述的等价性，并能精确刻画每半步的熵减少量。我们将此对偶性应用于若干具有实际意义的马尔可夫链，并针对对偶链的混合界获得了新结果或给出了简短的新证明，其中Swendsen-Wang动力学作为一个关键示例。此外，我们揭示了相关马尔可夫链的半步与源自熵正则化最优传输领域的Sinkhorn算法之间的相似性，二者均可统一于概率单纯形上关于α-散度的交替投影视角之下。