Breaking the Curse of Dimensionality

By Mark Coppejans. 2000

140 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In the context of numerical analysis, a decomposition of the form (5) was first proposed in [212, 213, 217].”
On manifolds of tensors of fixed TT-rank (2012, background): “Independently, the TT format (abbreviating “tree tensor” or “tensor train”), a special case of the above hierarchical HT structure, was recently introduced by Oseledets and Tyrtyshnikov [34, 38, 39].”
A New Truncation Strategy for the Higher-Order Singular Value Decomposition (2012, methodology): “It is a building block for other tensor decompositions, such as the hierarchical Tucker decomposition [18, 21], the Tucker tree decomposition [43, 44], and the tensor-train decomposition [42].”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, methodology): “Canonical and Tucker formats can be straightforwardly generalized to d dimensions, however each of them has serious drawbacks, and another decomposition should be used for high dimensions, for example recently introduced tensor-train (TT decomposition, see [28, 23, 24]), that are based on the SVD te”
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats (2014, background): “As mentioned for the tensor formats above, the numerical cost for operation with tensors represented in the TT-format is linear in K, see for example [12, 24, 25].”
Tensor-Train Ranks for Matrices and Their Inverses (2011, methodology): “Note that recursive approaches as a way to compact representations of multi-index arrays were first proposed in [6, 14, 15].”
The geometry of algorithms using hierarchical tensors (2013, background): “In this short section we show how the TT format of [14,17] is obtained as a special case of the HT format.”
Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs (2010, background): “Furthermore, the recently analysed dimension splitting TC/TT tensor formats can be considered [30, 28, 22].”
Tensor-Train Decomposition (2011, methodology): “As in [32], all ranks involved are equal to 2, and the tensor is represented by a set of (d−2) arrays of sizes 2×n×2 and two n×2 matrices.”
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 ”
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals (2014, methodology): “For problems in higher dimensions, the low-parametric rankstructured tensor representation of functions can be based on the commonly used matrix product states (MPS) [41,42] or equivalently tensor train (TT) [43] formats.”
Superfast Fourier Transform Using QTT Approximation (2012, methodology): “Recently proposed tensor train (TT) format [48, 43, 46], also known in quantum chemistry as matrix product states (MPS) [60], combines advances of both the canonical and Tucker formats: the number of representing parameters grows only linearly with the dimension and the approximation problem is stab”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, background): “, rd−1 are called TT ranks of the decomposition [33, 32].”
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation (2011, methodology): “In the recent mathematical literature the various versions of MPS tensor decomposition were discovered as the hierarchical Tucker [13], the tensor train (TT) [23] and the tensor chain (TC) [17] formats.”
Randomized Matrix Computations (2012, unknown): “Its previous study can be largely traced through the papers [GTZ97], [GT01], [GOS08], [T00], [MMD08], [OT09], [HMT11], and the bibliography therein, but much earlier advances in this area appeared in the papers [BCLR79], [B80], [B85], [B86], [BC87], directed to estimating the border rank of matrices”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, background): “Typical choices for such dimension trees are those obtained by a balanced splitting of the index set, as used, e.g., by Grasedyck [7], or a front-to-back splitting which (with minor adaptations; see Remark 1) leads to the TT format due to Oseledets and Tyrtyshnikov [25, 26].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “Furthermore, the SVD/QR-based algorithms for the rank optimization in the TT/HT dimension splitting schemes have been described in [98, 97, 79, 26].”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, background): “In the recent mathematical literature the various versions of matrix product tensor decomposition were discussed as the tensor train (TT) [36, 38], the tensor chain (TC) [26] and the closely related hierarchical Tucker [15] formats.”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, methodology): “In this paper Tensor Train (TT) and (Quantics Tensor Train) (QTT) representation will be utilized heavily both for matrices and vectors (for detailed description see [22, 24, 23, 15, 12]), and for fast linear algebra operations in these formats TT-Toolbox1 will be used.”