# Optimization On Manifolds: Methods And Applications

@inproceedings{Absil2010OptimizationOM, title={Optimization On Manifolds: Methods And Applications}, author={Pierre-Antoine Absil and Robert E. Mahony and Rodolphe Sepulchre}, year={2010} }

This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however,we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton’s method in some detail as well as…

## 31 Citations

A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds

- MathematicsJ. Optim. Theory Appl.
- 2016

Without requiring any restrictive assumptions about the sign of the sectional curvature of the manifold, full convergence is obtained for any bounded sequence generated by the proximal point method, in the case that the objective function satisfies the Kurdyka–Lojasiewicz inequality.

Optimization Methods on Riemannian Manifolds and Their Application to Shape Space

- MathematicsSIAM J. Optim.
- 2012

We extend the scope of analysis for linesearch optimization algorithms on (possibly infinite-dimensional) Riemannian manifolds to the convergence analysis of the BFGS quasi-Newton scheme and the…

Accelerated Algorithms for Convex and Non-Convex Optimization on Manifolds

- Computer ScienceArXiv
- 2020

The proposed method unifies insights from Nesterov's original idea for accelerating gradient descent algorithms with recent developments in optimization algorithms in Euclidean space and shows that when the objective function is convex, the algorithm provably converges to the optimum and leads to accelerated convergence.

Proximal Point Methods for Lipschitz Functions on Hadamard Manifolds: Scalar and Vectorial Cases

- MathematicsJ. Optim. Theory Appl.
- 2018

We study the convergence of exact and inexact versions of the proximal point method with a generalized regularization function in Hadamard manifolds for solving scalar and vectorial optimization…

An Efficient BFGS Algorithm for Riemannian Optimization

- Mathematics, Computer Science
- 2010

A convergence result for Riemannian line-search methods that ensures superlinear convergence is presented and a theory of building vector transports on submanifolds of R n is presented.

Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts

- Mathematics
- 2020

A suitable definition of a tangent space in view to optimization methods is presented and a diffeological Riemannian space and a Diffeological gradient, which are needed in an optimization algorithm on diffeology spaces are presented.

Projection-like Retractions on Matrix Manifolds

- MathematicsSIAM J. Optim.
- 2012

This theory offers a framework in which previously proposed retractions can be analyzed, as well as a toolbox for constructing new ones, for submanifolds of Euclidean spaces.

Piecewise rigid curve deformation via a Finsler steepest descent

- Mathematics
- 2013

This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al., to the setting of Finsler…

Curvature and Torsion estimation of 3D functional data: A geometric approach to build the mean shape under the Frenet Serret framework

- Computer Science, Mathematics
- 2022

This work develops an alternative characterization of a mean that reflects shape variation of the curves and introduces a new definition of mean curvature and mean torsion, as well as mean shape through the notion of mean vector field.

A Variational Approach to Registration with Local Exponential Coordinates

- Computer ScienceComput. Graph. Forum
- 2019

We identify a novel parameterization for the group of finite rotations (SO3), consisting of an atlas of exponential maps defined over local tangent planes, for the purpose of computing isometric…

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