Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Journal of Computational Physics
Linear algebra for tensor problems
Computing
Journal of Computational and Applied Mathematics
Global and fast receiver antenna selection for MIMO systems
IEEE Transactions on Communications
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors
SIAM Journal on Scientific Computing
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case
SIAM Journal on Scientific Computing
An equi-directional generalization of adaptive cross approximation for higher-order tensors
Applied Numerical Mathematics
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We consider Tucker-like approximations with an $r \times r \times r$ core tensor for three-dimensional $n \times n \times n$ arrays in the case of $r \ll n$ and possibly very large $n$ (up to $10^4$-$10^6$). As the approximation contains only $\mathcal{O}(rn + r^3)$ parameters, it is natural to ask if it can be computed using only a small amount of entries of the given array. A similar question for matrices (two-dimensional tensors) was asked and positively answered in [S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, A theory of pseudo-skeleton approximations, Linear Algebra Appl., 261 (1997), pp. 1-21]. In the present paper we extend the positive answer to the case of three-dimensional tensors. More specifically, it is shown that if the tensor admits a good Tucker approximation for some (small) rank $r$, then this approximation can be computed using only $\mathcal{O}(nr)$ entries with $\mathcal{O}(nr^{3})$ complexity.