Related papers: Faster Riemannian Newton-type Optimization by Subs…

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications. At the…

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical…

Hamiltonian simulation, i.e., simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful…

In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the…

One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformu-lation of the problem of interest and solving the resulting problem…

We focus on minimizing nonconvex finite-sum functions that typically arise in machine learning problems. In an attempt to solve this problem, the adaptive cubic regularized Newton method has shown its strong global convergence guarantees…

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle…

Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of…

We propose a Riemannian optimization approach for computing low-rank solutions of the algebraic Riccati equation. The scheme alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is on the set of…

We study optimization of finite sums of geodesically smooth functions on Riemannian manifolds. Although variance reduction techniques for optimizing finite-sums have witnessed tremendous attention in the recent years, existing work is…

In this paper, we propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for…

We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this…

In this paper, we propose a cubic regularized Newton (CRN) method for solving convex-concave saddle point problems (SPP). At each iteration, a cubic regularized saddle point subproblem is constructed and solved, which provides a search…

The trust-region (TR) method is renowned historically for its robustness in nonconvex problems and extraordinary numerical performance, but the study of its performance in convex optimization is somehow limited. This paper complements the…

High-dimensional data with intrinsic low-dimensional structure is ubiquitous in machine learning and data science. While various approaches allow one to learn a data manifold with a Riemannian structure from finite samples, performing…

This paper proposes a Riemannian adaptive optimization algorithm to optimize the parameters of deep neural networks. The algorithm is an extension of both AMSGrad in Euclidean space and RAMSGrad on a Riemannian manifold. The algorithm helps…

We study the performance of stochastic first-order methods for finding saddle points of convex-concave functions. A notorious challenge faced by such methods is that the gradients can grow arbitrarily large during optimization, which may…

Recently, the problem of local minima in very high dimensional non-convex optimization has been challenged and the problem of saddle points has been introduced. This paper introduces a dynamic type of normalization that forces the system to…

By restricting the iterate on a nonlinear manifold, the recently proposed Riemannian optimization methods prove to be both efficient and effective in low rank tensor completion problems. However, existing methods fail to exploit the easily…

In this paper, we study second-order algorithms for solving nonconvex-strongly concave minimax problems, which have attracted much attention in recent years in many fields, especially in machine learning.We propose a gradient norm…