We propose two approaches, based on Riemannian optimization, for computing a stochastic approximation of the $p$th root of a stochastic matrix $A$. In the first approach, the approximation is found in the Riemannian manifold of positive stochastic matrices. In the second approach, we introduce the Riemannian manifold of positive stochastic matrices sharing with $A$ the Perron eigenvector and we compute the approximation of the $p$th root of $A$ in such a manifold. This way, differently from the available methods based on constrained optimization, $A$ and its $p$th root approximation share the Perron eigenvector. Such a property is relevant, from a modelling point of view, in the embedding problem for Markov chains. The extended numerical experimentation shows that, in the first approach, the Riemannian optimization methods are generally faster and more accurate than the available methods based on constrained optimization. In the second approach, even though the stochastic approximation of the $p$th root is found in a smaller set, the approximation is generally more accurate than the one obtained by standard constrained optimization.

翻译：我们提出两种基于黎曼优化的方法，用于计算随机矩阵$A$的$p$次根的随机近似。在第一种方法中，近似解在正随机矩阵的黎曼流形上求得。在第二种方法中，我们引入与$A$共享Perron特征向量的正随机矩阵黎曼流形，并在该流形中计算$A$的$p$次根近似。这样，与基于约束优化的现有方法不同，$A$与其$p$次根近似共享Perron特征向量。从建模角度来看，这一性质在马尔可夫链的嵌入问题中具有重要意义。广泛数值实验表明，在第一种方法中，黎曼优化方法通常比基于约束优化的现有方法更快且更精确。在第二种方法中，尽管$p$次根的随机近似在更小的解集中求得，但其近似精度通常优于标准约束优化方法所得结果。