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Low rank Tucker-type tensor approximation to classical potentials

100%

Khoromskij B., Khoromskaia V.

Open Mathematics

2007

tom 5

nr 3

523-550

This paper investigates best rank-(r 1,..., r d) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r 1,..., r d) 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-(r 1,..., r d) 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.

Inertias and ranks of some Hermitian matrix functions with applications

81%

Zhang X., Wang Q.-W., Liu X.

Open Mathematics

2012

tom 10

nr 1

329-351

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of $$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$ The corresponding expressions of the maximal matrices of H are presented when the existence conditions are met. In this case, we further prove the matrix function f(X,Y)is invariant under changing the pair (X,Y). Moreover, we establish necessary and sufficient conditions for the system of matrix equations $$AX = B, YC = D, QXQ* + TYT* = P$$ to have a Hermitian solution and the system of matrix equations $$AX = C, BXB* = D$$ to have a bisymmetric solution. The explicit expressions of such solutions to the systems mentioned above are also provided. In addition, we discuss the range of inertias of the matrix functions P ± QXQ* ± TYT* where X and Y are a nonnegative definite pair of solutions to some consistent matrix equations. The findings of this pape extend some known results in the literature.

Ograniczanie wyników

2 Open Mathematics

1 Khoromskaia V.

1 Khoromskij B.

1 Liu X.

1 Wang Q.-W.

1 Zhang X.

1 2012

1 2007

Wyniki wyszukiwania

Low rank Tucker-type tensor approximation to classical potentials

Inertias and ranks of some Hermitian matrix functions with applications