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It was shown recently by Su et al. (2016) that Nesterov's accelerated gradient method for minimizing a smooth convex function $f$ can be thought of as the time discretization of a second-order ODE, and that $f(x(t))$ converges to its…
Copositive optimization is a special case of convex conic programming, and it consists of optimizing a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful…
Recently, a Riemannian proximal Newton method has been developed for optimizing problems in the form of $\min_{x\in\mathcal{M}} f(x) + \mu \|x\|_1$, where $\mathcal{M}$ is a compact embedded submanifold and $f(x)$ is smooth. Although this…
Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the…
Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure…
We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we…
Since the popularization of the Stiefel manifold for numerical applications in 1998 in a seminal paper from Edelman et al., it has been exhibited to be a key to solve many problems from optimization, statistics and machine learning. In…
Model reduction methods often aim at an identification of slow invariant manifolds in the state space of dynamical systems modeled by ordinary differential equations. We present a predictor corrector method for a fast solution of an…
We introduce a framework for molecular structure optimization using denoising model on a physics-informed Riemannian manifold (R-DM). Unlike conventional approaches operating in Euclidean space, our method leverages a Riemannian metric that…
We introduce a mathematical technique based on modifying a given Riemannian metric and we investigate its applicability to controlling and stabilizing constrained mechanical systems. In essence our result is based on the construction of a…
Local operations and classical communication (LOCC) is a foundational framework in quantum information from both theoretical and experimental perspectives. However, designing and optimizing LOCC protocols is intractable due to their complex…
Constant rescaling of a Riemannian metric appears in many computational settings, often through a global scale parameter that is introduced either explicitly or implicitly. Although this operation is elementary, its consequences are not…
Incorporating a non-Euclidean variable metric to first-order algorithms is known to bring enhancement. However, due to the lack of an optimal choice, such an enhancement appears significantly underestimated. In this work, we establish a…
This paper presents a memory efficient, first-order method for low multi-linear rank approximation of high-order, high-dimensional tensors. In our method, we exploit the second-order information of the cost function and the constraints to…
The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms…
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…
A novel preconditioner of Neumann-Neumann type for the Stokes-Darcy problem is studied, where optimal weights of the local subproblems that define the preconditioner are obtained by minimizing the convergence rate of the method in the…
The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two…
The paper proposes a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The…
The steepest descent method for multiobjective optimization on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The aim of the paper is twofold. Firstly, an asymptotic analysis of the method is…