Tensor and its tucker core: The invariance relationships

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.