Abstract
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 1le ple infty. 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.
Highlights
The spectral p-norm of a tensor generalizes the spectral p-norm of a matrix
Most of the methods to tackle the tensor spectral p-norm and nuclear p-norm in the literature have been heavily relying on matrix unfoldings, no matter in theory
We systematically study the tensor spectral p-norm and nuclear p-norm via the partition approach in [14]
Summary
Most of the methods to tackle the tensor spectral p-norm and nuclear p-norm in the literature have been heavily relying on matrix unfoldings, no matter in theory Such as approximation methods [15] and in practice such as tensor completion [5]. We prove that for the most general partition called arbitrary partition, the bounds of the tensor spectral p-norm and nuclear p-norm via subtensors can be established for any 1 ≤ p ≤ ∞. It generalizes and answers affirmatively the Li’s conjecture, which is the case p = 2 for a tensor partition.
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