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}
}
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References

SHOWING 1-10 OF 13 REFERENCES

Low-rank tensor completion by Riemannian optimization

A new algorithm is proposed that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank with particular attention to efficient implementation, which scales linearly in the size of the tensor.

Introduction to Smooth Manifolds

Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves…

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

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.

Algebraic Geometry: A First Course

1: Affine and Projective Varieties. 2: Regular Functions and Maps. 3: Cones, Projections, and More About Products. 4: Families and Parameter Spaces. 5: Ideals of Varieties, Irreducible Decomposition.…

Condition - The Geometry of Numerical Algorithms

This book puts special emphasis on the probabilistic analysis of numerical algorithms via the analysis of the corresponding condition, and contains a condition-based course on linear programming that fills a gap between the current elementary expositions of the subject based on the simplex method and those focusing on convex programming.

A literature survey of low‐rank tensor approximation techniques

This survey attempts to give a literature overview of current developments in low-rank tensor approximation, with an emphasis on function-related tensors.

Induction for secant varieties of Segre varieties

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is…

Tensors: Geometry and Applications

This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry.

The Condition Number of Join Decompositions

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.

Matrix Perturbation Theory

X is the vector space which acts in the n-dimensional (complex) vector space R.1.1 and is related to Varepsilon by the following inequality.