Optimization and Learning over Riemannian Manifolds

Learning over smooth nonlinear spaces has found wide applications. A principled approach for addressing such problems is to endow the search space with a Riemannian manifold geometry and numerical optimization can be performed intrinsically. Recent years have seen a surge of interest in leveraging Riemannian optimization for nonlinearly-constrained problems. This thesis investigates and improves on the existing algorithms for Riemannian optimization, with a focus on unified analysis frameworks and generic strategies. To this end, the first chapter systematically studies the choice of Riemannian geometries and their impacts on algorithmic convergence, on the manifold of positive definite matrices. The second chapter considers stochastic optimization on manifolds and proposes a unified framework for analyzing and improving the convergence of Riemannian variance reduction methods for nonconvex functions. The third chapter introduces a generic acceleration scheme based on the idea of extrapolation, which achieves optimal convergence rate asymptotically while being empirically efficient

Newton acceleration on manifolds identified by proximal-gradient methods

Proximal methods are known to identify the underlying substructure of nonsmooth optimization problems. Even more, in many interesting situations, the output of a proximity operator comes with its structure at no additional cost, and convergence is improved once it matches the structure of a minimizer. However, it is impossible in general to know whether the current structure is final or not; such highly valuable information has to be exploited adaptively. To do so, we place ourselves in the case where a proximal gradient method can identify manifolds of differentiability of the nonsmooth objective. Leveraging this manifold identification, we show that Riemannian Newton-like methods can be intertwined with the proximal gradient steps to drastically boost the convergence. We prove the superlinear convergence of the algorithm when solving some nondegenerated nonsmooth nonconvex optimization problems. We provide numerical illustrations on optimization problems regularized by $\ell_1$-norm or trace-norm

Riemannian Optimization for Convex and Non-Convex Signal Processing and Machine Learning Applications

The performance of most algorithms for signal processing and machine learning applications highly depends on the underlying optimization algorithms. Multiple techniques have been proposed for solving convex and non-convex problems such as interior-point methods and semidefinite programming. However, it is well known that these algorithms are not ideally suited for large-scale optimization with a high number of variables and/or constraints. This thesis exploits a novel optimization method, known as Riemannian optimization, for efficiently solving convex and non-convex problems with signal processing and machine learning applications. Unlike most optimization techniques whose complexities increase with the number of constraints, Riemannian methods smartly exploit the structure of the search space, a.k.a., the set of feasible solutions, to reduce the embedded dimension and efficiently solve optimization problems in a reasonable time. However, such efficiency comes at the expense of universality as the geometry of each manifold needs to be investigated individually. This thesis explains the steps of designing first and second-order Riemannian optimization methods for smooth matrix manifolds through the study and design of optimization algorithms for various applications. In particular, the paper is interested in contemporary applications in signal processing and machine learning, such as community detection, graph-based clustering, phase retrieval, and indoor and outdoor location determination. Simulation results are provided to attest to the efficiency of the proposed methods against popular generic and specialized solvers for each of the above applications

Smoothing algorithms for nonsmooth and nonconvex minimization over the stiefel manifold

We consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with an linear mapping. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.Comment: 22 page