A New Scheme for the Tensor Representation

By Wolfgang Hackbusch, S. Kühn. 2009

532 citations recorded by the source corpus.

The paper presents a new scheme for the representation of tensors which is well-suited for high-order tensors. The construction is based on a hierarchy of tensor product subspaces spanned by orthonormal bases. The underlying binary tree structure makes it possible to apply standard Linear Algebra tools for performing arithmetical operations and for the computation of data-sparse approximations. In particular, a truncation algorithm can be implemented which is based on the standard matrix singular value decomposition (SVD) method.

Subfields: Artificial Intelligence, Computational Mathematics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In fact, a sparse grid approximation to f can be turned quite effectively into a low-rank tensor decomposition [108, 117].”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, background): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: High-dimensional problems, quantics folding of vectors, rank structured tensor approximation, matrix valued functions, FEM, material sciences, stochastic modeling.”
Optimization problems in contracted tensor networks (2011, background): “The most recent tensor representations are tensor networks, e.g. hierarchical tensor format [8, Hackbusch and Kühn, 2009], [6, Grasedyck, 2010], the tree Tucker format (TT) [18, 14, Oseledets and Tyrtyshnikov, 2009], where the TT tensor format is also called tensor train format.”
Fast adaptive interpolation of multi-dimensional arrays in tensor train format (2011, methodology): “Recently, for the H-Tucker format [26], [27], an adaptive black-box version based on the successive cross approximation of subtensors was presented by Ballani, Grasedyck and Kluge, with good examples of efficiency [28].”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “A more general format is the so-called HT format [16], but the TT format has a simpler algebraic structure and it is often easier to implement efficient methods in the TT format.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Generalized Cross Approximation for 3d-tensors Generalized Cross Approximation for 3d-tensors (background): “For the latest developements in this area, one can look at [3] [6] [7] [8] [11].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “The rst attempt used the hierarchical splitting of dimensions (see the hierarchical-type dimension splitting in [20], the H-Tucker format [12, 10] and the Tree-Tucker format [43, 38]), which are free from the curse of dimensionality, and what is important, basic MLA operations can be implemented in ”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “Hackbusch therefore generalized the idea of subspace approximation, being the basis of the Tucker decomposition, to a hierarchical ansatz proposed in [18]: High-dimensional tensors are therein successively decomposed to component tensors of order 3, which then can be treated separately, so that the ”
A projector-splitting integrator for dynamical low-rank approximation (2013, background): “Very recently, in [1, 9, 15] the dynamical low-rank approximation approach was extended to tensors in the tensor train (TT) format studied in [14] and the hierarchical Tucker (HT) format studied in [3].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “Another alternative to the Tucker format might be the HT format [44].”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “them are the Hierarchical (Tree) Tucker [21], or ML-MCTDH [20],”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, unknown): “Their dimensions are given by dimLi = ri−1, dimRi = ri (81) and hence, the full tensor |Ψ〉 is contained in the ri−1qiri-dimensional subspace(77,84) |Ψ〉 ∈ Li ⊗ Λi ⊗Ri ∼= Cri−1×qi×ri (82) |Ψ〉 = ∑”