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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 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 ϵ-stationary point is analyzed under mild assumptions. Existing ADMMs for solving nonconvex problems either do not allow a nonconvex constraint set or do not allow a nonsmooth objective function. Our algorithm is the first ADMM-type algorithm that minimizes a nonsmooth objective over manifold—a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method. Funding: The research of S. Ma was supported in part by the Office of Naval Research [Grant N00014-24-1-2705]; the National Science Foundation [Grants DMS-2243650, CCF-2308597, CCF-2311275, and ECCS-2326591]; the University of California, Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program; and Rice University start-up fund. 

We consider stochastic zeroth-order optimization over Riemannian submanifolds embedded in Euclidean space, where the task is to solve Riemannian optimization problems with only noisy objective function evaluations. Toward this, our main contribution is to propose estimators of the Riemannian gradient and Hessian from noisy objective function evaluations, based on a Riemannian version of the Gaussian smoothing technique. The proposed estimators overcome the difficulty of nonlinearity of the manifold constraint and issues that arise in using Euclidean Gaussian smoothing techniques when the function is defined only over the manifold. We use the proposed estimators to solve Riemannian optimization problems in the following settings for the objective function: (i) stochastic and gradient-Lipschitz (in both nonconvex and geodesic convex settings), (ii) sum of gradient-Lipschitz and nonsmooth functions, and (iii) Hessian-Lipschitz. For these settings, we analyze the oracle complexity of our algorithms to obtain appropriately defined notions of ϵ-stationary point or ϵ-approximate local minimizer. Notably, our complexities are independent of the dimension of the ambient Euclidean space and depend only on the intrinsic dimension of the manifold under consideration. We demonstrate the applicability of our algorithms by simulation results and real-world applications on black-box stiffness control for robotics and black-box attacks to neural networks.