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.
Bibliographical noteFunding Information:
We are very grateful to the two anonymous referees for their invaluable suggestions, which helped to improve this paper from its original version. We would like to thank Chunfeng Cui for sharing with us the codes on computing all Z-eigenvalues, and we thank Shmuel Friedland, Lek-Heng Lim, Jiawang Nie, and Nikos Sidiropoulos for the fruitful discussions on the topics related to this paper. The research of Bo Jiang was supported in part by National Natural Science Foundation of China (Grant 11401364) and Program for Innovative Research Team of Shanghai University of Finance and Economics. The research of Shuzhong Zhang was supported in part by National Science Foundation (Grant CMMI-1462408).
- CP decomposition
- Tucker decomposition
- border rank
- tensor Schatten quasi norm
- tensor eigenvalues