Stochastic \({p}\)th Root Approximation of a Stochastic Matrix: A Riemannian Optimization Approach

We propose two approaches, based on Riemannian optimization for computing a stochastic approximation of the th root of a stochastic matrix . 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 the Perron eigenvector and we compute the approximation of the th root of in such a manifold. This way, differently from the available methods based on constrained optimization, and its th root approximation share the Perron eigenvector. Such a property is relevant, from a modeling 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 th root is found in a smaller set, the approximation is generally more accurate than the one obtained by standard constrained optimization.