f¨ur Mathematik in den Naturwissenschaften Leipzig Low Rank Tucker-Type Tensor Approximation to Classical Potentials

By Boris N. Khoromskij, Venera Khoromskaia

83 citations recorded by the source corpus.

AbstractThis paper investigates best rank-(r1,..., rd) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r1,..., rd) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r1,..., rd) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example $$\tfrac{1}{{\left| {x - y} \right|}}, e^{ - \alpha \left| {x - y} \right|} , \tfrac{{e^{ - \left| {x - y} \right|} }}{{\left| {x - y} \right|}}$$ and $$\tfrac{{erf(|x|)}}{{|x|}}$$ with x, y ∈ ℝd. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…[152]; • computational tasks in electronic structure calculations, e.g., based on Hartree-Fock or DFT models [23,24,31,44,45,47,87,141–146,150,156–159]; • the solution of stochastic and parametric PDEs [68,69,76,77,161,165,171,189,202]; • approximation of Green’s functions of…”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, background): “Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the curse of dimensionality [2, 3, 24, 25, 33].”
Verification of the cross 3D algorithm on quantum chemistry data (2008, background): “A detailed discussion of the two-level tensor representations (formats) can be found in [16, 18 , 19].”
Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd (2009, methodology): “Applications of tensor methods for representation of classical Green’s kernels and elliptic resolvent [3, 4, 11, 13, 19, 23 , 2], multi-dimensional convolution [21, 20, 6] and many other quantities arising in electronic structure calculations [17, 22, 24] have demonstrated the surprising efficiency”
Use of tensor formats in elliptic eigenvalue problems (2012, unknown): “decay of ((w,MR) with respect to R. For its computation and analysis see [12, 11, 23, 14, 17 , 15, 4].”
Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors (2008, methodology): “It can be used for approximating higherorder tensors, obtained by discretizing Newton potential, Slater-type functions, Yukawa and Helmholtz potentials [31].”
Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels (2012, methodology): “To construct tensor-structured representations, we make use of the well-known Gaussian transform of the potentials, and then approximate the resulting univariate integral in R by special sinc quadratures.”
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors (2009, background): “In part to illustrate the efficiency of these algorithms, this paper also addresses the following two related problems about the multilinear approximations of tensors and symmetric tensors, which are also important problems in their own right with various applications in analytical chemistry [52], b”
A New Scheme for the Tensor Representation (2009, background): “partial data is solved differently in [4] and [8].”
Tensor Decompositions and Applications (2009, methodology): “Khoromskij and Khoromskaia [115] have applied CANDELINC (which they call “Tucker-to-canonical”) to approximations of classical potentials.”