Convergence Results for Projected Line-Search Methods on Varieties of Low-Rank Matrices Via Łojasiewicz Inequality
- R. Schneider, A. Uschmajew
- MathematicsSIAM Journal on Optimization
- 21 February 2014
Convergence results for projected line-search methods on the real-algebraic variety $\mathcal{M}_{\le k}$ of real $m \times n$ matrices of rank at most $k$ are derived.
Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation
- A. Uschmajew
- MathematicsSIAM Journal on Matrix Analysis and Applications
- 28 June 2012
A local convergence theorem for calculating canonical low-rank tensor approximations (PARAFAC, CANDECOMP) by the alternating least squares algorithm is established. The main assumption is that the…
The geometry of algorithms using hierarchical tensors
- A. Uschmajew, Bart Vandereycken
- MathematicsLinear Algebra and its Applications
- 1 July 2013
Parallel algorithms for tensor completion in the CP format
- L. Karlsson, D. Kressner, A. Uschmajew
- Computer ScienceParallel Computing
- 1 September 2016
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format
- T. Rohwedder, A. Uschmajew
- Computer ScienceSIAM Journal on Numerical Analysis
- 4 April 2013
The positive practical experience with TT-ALS is backed up with an according local linear convergence theory for the optimization of convex functionals $J$ and the main assumption entering the proof is that the redundancy introduced by the TT parametrization matches the null space of the Hessian of the induced functional.
Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems
- D. Kressner, M. Steinlechner, A. Uschmajew
- Computer ScienceSIAM Journal on Scientific Computing
- 14 October 2014
This work considers the solution of large-scale symmetric eigenvalue problems for which it is known that the eigenvectors admit a low-rank tensor approximation from the discretization of high-dimensional elliptic PDE eigen value problems or in strongly correlated spin systems.
A new convergence proof for the higher-order power method and generalizations
- A. Uschmajew
- Mathematics
- 17 July 2014
A proof for the point-wise convergence of the factors in the higher-order power method for tensors towards a critical point is given. It is obtained by applying established results from the theory of…
Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
- M. Bachmayr, R. Schneider, A. Uschmajew
- Computer Science, MathematicsFoundations of Computational Mathematics
- 1 December 2016
A survey of developments of techniques for the computation of hierarchical low-rank approximations, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations.
A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds
- S. Hosseini, A. Uschmajew
- MathematicsSIAM Journal on Optimization
- 8 February 2017
The method is based on approximating the subdifferential of the cost function at every iteration by the convex hull of transported gradients from tangent spaces at randomly generated nearby points to the tangent space of the current iterate and can hence be seen as a generalization of the well known gradient sampling algorithm to a Riemannian setting.
Greedy rank updates combined with Riemannian descent methods for low-rank optimization
- A. Uschmajew, Bart Vandereycken
- Computer ScienceInternational Conference on Sampling Theory and…
- 25 May 2015
A rank-adaptive optimization strategy for finding low-rank solutions of matrix optimization problems involving a quadratic objective function that can be interpreted as a perturbed gradient descent algorithms or as a simple warm-starting strategy of a projected gradient algorithm on the variety of matrices of bounded rank.
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