Tensor-Train Decomposition
Ivan Oseledets
Abstract
A simple nonrecursive form of the tensor decomposition in d dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.
Topics & Concepts
MathematicsComputationCurse of dimensionalityDecompositionEigenvalues and eigenvectorsTensor algebraEigendecomposition of a matrixTensor (intrinsic definition)Simple (philosophy)RoundingRank (graph theory)Algebra over a fieldLinear algebraOperator (biology)Applied mathematicsMatrix decompositionAlgorithmPure mathematicsComputer scienceCombinatoricsAlgebra representationGeometryJordan algebraRepressorBiologyBiochemistryChemistryEcologyStatisticsGeneOperating systemEpistemologyTranscription factorQuantum mechanicsPhysicsPhilosophyTensor decomposition and applicationsMatrix Theory and AlgorithmsElectromagnetic Scattering and Analysis