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$p$次根的随机逼近。第一种方法中，逼近在正随机矩阵的黎曼流形上求解；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，并在该流形上计算$A$的$p$次根逼近。这种方式与现有的基于约束优化的方法不同，使得$A$及其$p$次根逼近共享Perron特征向量。从建模角度看，该性质对马尔可夫链的嵌入问题具有重要意义。广泛的数值实验表明：在第一种方法中，黎曼优化方法通常比现有基于约束优化的方法更快且更精确；在第二种方法中，尽管$p$次根的随机逼近在更小的集合中求解，但其逼近精度通常优于标准约束优化方法。