# Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number

@article{Breiding2017ConvergenceAO, title={Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number}, author={Paul Breiding and Nick Vannieuwenhoven}, journal={Appl. Math. Lett.}, year={2017}, volume={78}, pages={42-50} }

## 15 Citations

### Spectral residual method for nonlinear equations on Riemannian manifolds

- MathematicsComputational and Applied Mathematics
- 2021

Numerical results indicate that the spectral algorithm for nonlinear equations (SANE) is very effective and efficient solving tangent vector field on different Riemannian manifolds and competes favorably with a Polak-Ribi\'ere-Polyak Method recently published and other methods existing in the literature.

### The Condition Number of Riemannian Approximation Problems

- MathematicsSIAM J. Optim.
- 2021

The main results are validated through experiments with the $n$-camera triangulation problem in computer vision and the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem is characterized.

### Riemannian Levenberg-Marquardt Method with Global and Local Convergence Properties

- Mathematics
- 2022

We extend the Levenberg-Marquardt method on Euclidean spaces to Riemannian manifolds. Although a Riemannian Levenberg–Marquardt (RLM) method was proposed by Peeters in 1993, to the best of our…

### Learning Paths from Signature Tensors

- Computer Science, MathematicsSIAM J. Matrix Anal. Appl.
- 2019

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry, and numerical optimization to this group action. Given a tensor in the...

### Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence

- Computer ScienceArXiv
- 2021

It is proved the first local quadratic convergence guarantee of RGN for low-rank tensor estimation in the noisy setting under some regularity conditions and the corresponding estimation error upper bounds are provided.

### Towards a condition number theorem for the tensor rank decomposition

- Mathematics
- 2018

We show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions (i.e., decompositions with unbounded geometric condition number, derived in [P.…

### A theory of condition for unconstrained perturbations

- Computer ScienceArXiv
- 2019

It is shown that the way how the input space is curved inside the ambient space affects the condition number, and a connection to the sensitivity analysis of Riemannian optimization problems is exhibited.

### The Dynamics of Swamps in the Canonical Tensor Approximation Problem

- Computer ScienceSIAM J. Appl. Dyn. Syst.
- 2019

The ability to approximate a multivariate function/tensor as a sum of separable functions/tensors is quite useful. Unfortunately, optimization-based algorithms to do so regularly exhibit unusual tr...

### A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

- Computer Science, MathematicsSIAM J. Optim.
- 2018

A Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs and a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions.

### THE CONDITION NUMBER OF JOIN

- Mathematics, Computer Science
- 2017

This paper examines the numerical sensitivity of join decompositions to perturbations and proves that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix.

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A Riemannian Gauss-Newton method with trust region for solving small-scale, dense TAPs and a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and which often yields a quick escape path from such spurious decompositions.

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This paper examines the numerical sensitivity of join decompositions to perturbations and proves that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix.

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