Reviews: Inexact trust-region algorithms on Riemannian manifolds

This paper proposed the inexact TR optimization algorithm on manifold, in which the analysis largely follows from the euclidean case [27,28]. They also considered the subsampling variant of their method for finite-sum problems and the empirical performance on PCA and matrix completion problems. It seems to me the convergence analysis follows exactly from the works [27,28] except for some different notation. At least this paper doesn't show what is the technical challenge for the Riemannian case. What are the advantages of using Riemannian optimization method?