On the tensor spectral p-norm and its dual norm via partitions

This paper presents a generalization of the spectral norm and the nuclear norm of a tensor via arbitrary tensor partitions, a much richer concept than block tensors. We show that the spectral p-norm and the nuclear p-norm of a tensor can be lower and upper bounded by manipulating the spectral p-norms and the nuclear p-norms of subtensors in an arbitrary partition of the tensor for 1≤p≤∞. Hence, it generalizes and answers affirmatively the conjecture proposed by Li (SIAM J Matrix Anal Appl, 37:1440-1452, 2016) for a tensor partition and p=2. We study the relations of the norms of a tensor, the norms of matrix unfoldings of the tensor, and the bounds via the norms of matrix slices of the tensor. Various bounds of the tensor spectral and nuclear norms in the literature are implied by our results.