André Uschmajew

Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation

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 Hessian matrix of the problem is positive definite modulo the scaling indeterminacy. A discussion, whether this is realistic, and numerical illustrations are included. Also regularization is addressed.

On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format

Alternating linear schemes (ALS), with the alternating least squares algorithm a notable special case, provide one of the simplest and most popular choices for the treatment of optimization tasks by tensor methods. An according adaptation of ALS for the recent TT (= tensor train) format [I. V. Oseledets, SIAM J. Sci. Comput., 33 (2011), pp. 2295--2317], known in quantum computations as matrix product states, has recently been investigated in [S. Holtz, T. Rohwedder, and R. Schneider, SIAM J. Sci. Comput., 34 (2012), pp. A683--A713]. With the present work, the positive practical experience with TT-ALS is backed up with an according local linear convergence theory for the optimization of convex functionals

Convergence Results for Projected Line-Search Methods on Varieties of Low-Rank Matrices Via Łojasiewicz Inequality

J

Low-Rank Tensor Methods with Subspace Correction for Symmetric Eigenvalue Problems

. The main assumption entering the proof is that the redundancy introduced by the TT parametrization

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

\tau

On convergence of the maximum block improvement method

matches the null space of the Hessian of the induced functional

Parallel algorithms for tensor completion in the CP format

j = J \circ \tau

Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations

, and we give conditions under which this assumption can be expected to hold. In particular, this is the case if ...

A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety

Greedy rank updates combined with Riemannian descent methods for low-rank optimization

\mathcal{M}_{\le k}