A concise proof to the spectral and nuclear norm bounds through tensor partitions

Xu Kong

doi:10.1515/math-2019-0028

Abstract

Abstract On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

Highlights

Tensor is the main subject in multilinear algebra [1,2,3,4,5,6]
A tensor T = ∈ Rn ×n ×···×nd is a d-way array, i.e., its entries ti i ···id are represented via d indices, say i, i, · · ·, id with each index ranging from to nj, ≤ j ≤ d
In [15], Li proposed an e cient way for the estimation of the tensor spectral and nuclear norms based on tensor partitions, which is de ned as follows

Summary

Introduction

Tensor is the main subject in multilinear algebra [1,2,3,4,5,6]. Let R be the eld of real numbers. In [15], Li proposed an e cient way for the estimation of the tensor spectral and nuclear norms based on tensor partitions, which is de ned as follows. Let X ∈ Rn ×n ×···×nd and {X , X , · · · , Xm} be a general tensor partition of X.

Results

Conclusion