Related papers: Generalized Left-Localized Cayley Parametrization …
Orthogonality constraints naturally appear in many machine learning problems, from principal component analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the…
This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an…
In Riemannian computing applications, it is crucial to map manifold data to a Euclidean domain, where vector space arithmetic is available, and back. Classical manifold theory guarantees the existence of such mappings, called charts and…
Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…
Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices…
The problem of optimization on Stiefel manifold, i.e., minimizing functions of (not necessarily square) matrices that satisfy orthogonality constraints, has been extensively studied. Yet, a new approach is proposed based on, for the first…
A novel algorithm for designing optimized orthonormal transform-matrix codebooks for adaptive transform coding of a non-stationary vector process is proposed. This algorithm relies on a block-wise stationary model of a non-stationary…
Many classical and modern machine learning algorithms require solving optimization tasks under orthogonality constraints. Solving these tasks with feasible methods requires a gradient descent update followed by a retraction operation on the…
We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the…
We introduce a manifold-based framework for addressing optimization problems with equality and inequality constraints found in robotics. Our approach transforms the original problem into an unconstrained optimization problem directly on the…
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic…
We develop a generalization of Nesterov's accelerated gradient descent method which is designed to deal with orthogonality constraints. To demonstrate the effectiveness of our method, we perform numerical experiments which demonstrate that…
Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…
We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which the objective function is weakly convex in the ambient Euclidean space. Such problems are ubiquitous in engineering applications but still largely…
This paper describes a general formalism for obtaining localized solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems. This class includes the important cases of Schr\"odinger's…
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and…
Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem…
We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of problems can be classified into three classes.…
We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose…
In this paper, we propose a general algorithmic framework to solve a class of optimization problems on the product of complex Stiefel manifolds based on the matrix polar decomposition. We establish the weak convergence, global convergence…