  We consider a class of Riemannian optimization problems where the objective
is the sum of a smooth function and a nonsmooth function, considered in the
ambient space. This class of problems finds important applications in machine
learning and statistics such as the sparse principal component analysis, sparse
spectral clustering, and orthogonal dictionary learning. We propose a
Riemannian alternating direction method of multipliers (ADMM) to solve this
class of problems. Our algorithm adopts easily computable steps in each
iteration. The iteration complexity of the proposed algorithm for obtaining an
$\epsilon$-stationary point is analyzed under mild assumptions. Existing ADMM
for solving nonconvex problems either does not allow nonconvex constraint set,
or does not allow nonsmooth objective function. In contrast, our complexity
result is established for problems with simultaneous nonsmooth objective and
manifold constraint. Numerical experiments are conducted to demonstrate the
advantage of the proposed method.
