A Riemannian ADMM

Optimization over Riemannian manifolds has drawn a lot of attention due to its applications in machine learning and related disciplines, including low-rank matrix completion [6, 49], phase retrieval [3, 45], blind deconvolution [21] and dictionary learning [11, 43]. Riemannian optimization aims at minimizing an objective function over a Riemannian manifold. When the objective function is smooth, people have proposed to solve them using Riemannian gradient method, Riemannian quasi-Newton method, Riemannian trust-region method, etc. Work along this line has been summarized in the monographs [1, 5] as well as some other references. Recently, due to increasing demand from application areas such as machine learning, statistics, signal processing and so on, there is a line of work designing efficient and scalable algorithms for solving Riemannian optimization problems with nonsmooth objectives. For example, people have studied Riemannian subgradient method [33], Riemannian proximal gradient method [10, 23], Riemannian proximal point algorithm [9], Riemannian proximal-linear algorithm [51], zeroth-order Riemannian algorithms [32], and so on. One thing that has not been widely considered is how to design alternating direction method of multipliers (ADMM) on manifolds.