Tensor approximations of matrices generated by asymptotically smooth functions

By E. E. Tyrtyshnikov. 2003

106 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Numerical Analysis

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other techniques include applying numerical quadrature to an integral representation of the function, see, e.g., [24, 77, 87, 110, 115], Taylor series expansion [249, 250], and polynomial interpolation [19, 33].”
Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators (2005, unknown): “[1, 17, 20]).”
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time (2008, background): “The rank estimates obtained in [15, 19, 20] have form r 6 C(log n log(2) ε), where ε is an error of the approximation, so the rank grows only logarithmically with n and ε.”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, background): “[83, 84]) it is possible to show that tensor-product approximations with low separation rank exist.”
Tensor-Train Ranks for Matrices and Their Inverses (2011, background): “For example, this was behind the numerical examples in [21].”
FFT-based Kronecker product approximation to micromagnetic long-range interactions (2012, background): “For matrices A ∈ R 3 i = 1 n i × (cid:81) 3 i = 1 n i , typically arising from discretized operators, the canonical format is usually given in Kronecker product form [31] with matrices U ( q ) j ∈ R n q × n q , scalars α j and ⊗ equals Kronecker product .”
Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part I. Separable Approximation of Multi-variate Functions (2005, methodology): “, ζd) := r ∑ k=1 (1)k(ζ1) · · · dk (ζd) ≈ F, where the set of univariate functions { k(·) : 1 ≤ ≤ d, 1 ≤ k ≤ r} may be fixed or chosen adaptively (see the discussion in [1, 15, 19]).”
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions (2009, background): “This kind of approximation exists for many important cases, for example, for asymptotically smooth functions [17] such as the kernels of important integral operators.”
Tensor-Train Decomposition (2011, background): “” Such permutation of dimensions is standard in the compression of high-dimensional operators [37, 17, 16].”
Algorithms for Numerical Analysis in High Dimensions (2005, background): “The class of complete asymptotically smooth functions is considered in [16, 43, 42].”
QTT approximation of elliptic solution operators in higher dimensions (2011, methodology): “The Quantics-TT (QTT) format [21, 32, 33] combines the idea of introducing additional ‘virtual levels’ (such idea was first proposed in [37]) with the tensor-train decomposition.”
TT-cross approximation for multidimensional arrays (2010, background): “For several known classes of tensors (arising, for example, from discretization of integral or diﬀerential operators), by analytical techniques one can prove the existence of low-tensor-rank approximations to A ( i 1 , i 2 , . . . , i d ) (cf. [40,26,28,10,5,6,25]).”
QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images (2012, background): “Suprisingly, this format can be applied also to data of low dimension using the virtual levels [16] / quantization of indices [9].”
Tensor-Product Approximation to Multidimensional Integral Operators and Green's Functions (2008, background): “[1, 19, 28]) to reach the complexity O(dn log n) with q ≥ 1 independent of d (see section 2.”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, background): “The use of quantization in the context of tensor decompositions dates back to [54].”