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Tensor and its tucker core: The invariance relationships
Author(s) -
Jiang Bo,
Yang Fan,
Zhang Shuzhong
Publication year - 2017
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2086
Subject(s) - tensor (intrinsic definition) , mathematics , multilinear algebra , tucker decomposition , symmetric tensor , tensor contraction , rank (graph theory) , multilinear map , eigenvalues and eigenvectors , curse of dimensionality , invariants of tensors , tensor algebra , algebra over a field , core (optical fiber) , lanczos tensor , polynomial , cartesian tensor , pure mathematics , tensor density , combinatorics , tensor field , exact solutions in general relativity , mathematical analysis , tensor product , tensor decomposition , computer science , algebra representation , jordan algebra , physics , quantum mechanics , telecommunications , statistics
Summary In one study, Hillar and Lim famously demonstrated that “multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are nondeterministic polynomial‐time hard.” Despite many recent advancements, the state‐of‐the‐art methods for computing such “tensor analogues” still suffer severely from the curse of dimensionality. In this paper, we show that the Tucker core of a tensor, however, retains many properties of the original tensor, including the CANDECOMP/PARAFAC (CP) rank, the border rank, the tensor Schatten quasi norms, and the Z‐eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker‐like decomposition that can accommodate any full column‐rank factor matrices.