The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format

By Sebastian Holtz, Thorsten Rohwedder, Reinhold Schneider. 2012

421 citations recorded by the source corpus.

Subfields: Computational Mathematics, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

In Electrical and Electronic Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Later on, the ideas of DMRG have been picked up and extended to other applications in the numerical analysis community in a series of papers [65,124,127,160,171,206].”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, methodology): “Our practical experience, as partly reported in [22], is that the ALS shows extraordinarily nice convergence behavior similar to that of ALS applied to the Tucker format.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, positive): “4 Orthogonalization of Tensor Train Network The orthogonalization of core tensors is an essential procedure in many algorithms for the TT formats (Oseledets, 2011; Holtz et al., 2012; Dolgov et al., 2014; Dolgov, 2014; Kressner et al., 2014a; Steinlechner, 2015, 2016).”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, background): “The general ALS and DMRG schemes for the typical quadratic functionals (error, residual, Rayleigh quotient) were discussed in [19], and the QTT-DMRG eigenvalue solver was presented in [28].”
Algorithm 941 (2014, methodology): “In the following, we briefly introduce tensor network diagrams (also called Penrose diagrams) to conveniently describe algorithms for contractions of tensors in HTD; see also Holtz et al. [2012] and Huckle et al.”
The geometry of algorithms using hierarchical tensors (2013, methodology): “In contrast to [31], we also establish a globally smooth structure by employing the standard construction of a smooth Lie group action.”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “The TT-DMRG ([16], MALS in [39], TT-solve in [42], DMRG MatVec in [51]) allows to determine the ranks adaptively at runtime.”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, methodology): “The second approach [23, 24, 25, 26, 27] are the DMRG or (M)ALS methods, which work directly within the structure of the low rank tensor format, typically the hierarchical Tucker or the Tensor Train format.”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, background): “More on the mathematical ideas behind the ALS and DMRG optimization in the TT format can be found in [76].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “In the following, we briefly introduce the concept of tensor network diagrams [35, 36].”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “It was later applied [27, 23] to solve an SPD linear system Ax = b by the minimization of the energy function (1.”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, methodology): “Approximately computing the dynamical approximation to A+ tΔA with one or few Euler steps provides an alternative to truncation steps by higher order SVD [7], or by local optimization methods like alternating least squares methods [11].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “It employs the ideas of the alternating least squares (ALS) and MALS (DMRG) methods [26, 27, 28, 29], improved by combining with the GMRES approach to determine a new correction direction.”
Towards an ADI iteration for Tensor Structured Equations (2014, methodology): “We have, however to note, that our numerical examples show that it is more efficient to apply a DMRG (density matrix renormalization group) solver [40], see [10, 16] for a description of the used DMRG solver for tensor-trains.”