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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. 

Spectral clustering is one of the fundamental unsupervised learning methods and is widely used in data analysis. Sparse spectral clustering (SSC) imposes sparsity to the spectral clustering, and it improves the interpretability of the model. One widely adopted model for SSC in the literature is an optimization problem over the Stiefel manifold with nonsmooth and nonconvex objective. Such an optimization problem is very challenging to solve. Existing methods usually solve its convex relaxation or need to smooth its nonsmooth objective using certain smoothing techniques. Therefore, they were not targeting solving the original formulation of SSC. In this paper, we propose a manifold proximal linear method (ManPL) that solves the original SSC formulation without twisting the model. We also extend the algorithm to solve multiple-kernel SSC problems, for which an alternating ManPL algorithm is proposed. Convergence and iteration complexity results of the proposed methods are established. We demonstrate the advantage of our proposed methods over existing methods via clustering of several data sets, including University of California Irvine and single-cell RNA sequencing data sets.