相关论文: Constrained dynamics for searching saddle points o…

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified…

This paper focuses on the distributed optimization of stochastic saddle point problems. The first part of the paper is devoted to lower bounds for the centralized and decentralized distributed methods for smooth (strongly) convex-(strongly)…

In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties:…

An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The…

The paper studies a distributed gradient descent (DGD) process and considers the problem of showing that in nonconvex optimization problems, DGD typically converges to local minima rather than saddle points. The paper considers…

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations…

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive…

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…

This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization of the solutions of the linear program as saddle points of a modified Lagrangian function. We show…

Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this…

A Riemannian stochastic representation of model uncertainties in molecular dynamics is proposed. The approach relies on a reduced-order model, the projection basis of which is randomized on a subset of the Stiefel manifold characterized by…

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…

In part I we considered the problem of convergence to a saddle point of a concave-convex function via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of…

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…

Stochastic contraction analysis is a recently developed tool for studying the global stability properties of nonlinear stochastic systems, based on a differential analysis of convergence in an appropriate metric. To date, stochastic…

Despite decades of work in fast reactive planning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This…

State-space smoothing has found many applications in science and engineering. Under linear and Gaussian assumptions, smoothed estimates can be obtained using efficient recursions, for example Rauch-Tung-Striebel and Mayne-Fraser algorithms.…

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…

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian…

Excited electronic states of molecules and solids play a fundamental role in fields such as catalysis and electronics. In electronic structure calculations, excited states typically correspond to saddle points on the surface described by…