Relations of the Nuclear Norms of a Tensor and its Matrix Flattenings

TL;DR

This paper establishes bounds relating the nuclear norms of a 3-tensor and its matrix flattenings, providing sharp bounds and a criterion for tightness, with generalizations to higher-order tensors.

Contribution

It introduces bounds connecting tensor nuclear norms with matrix flattenings and offers a criterion for when the bounds are tight, extending to N-tensors.

Findings

Nuclear norm of tensor flattenings lower bounds the tensor nuclear norm.

Tensor nuclear norm is upper bounded by a factor times flattening nuclear norms.

Bounds are sharp for 3-tensors, with a criterion for tightness provided.

Abstract

For a $3$-tensor of dimensions $I_1\times I_2\times I_3$, we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm, and which in turn is upper bounded by $\sqrt{\min\{I_i : i\neq j\}}$ times the nuclear norm of the matrix flattening in mode $j$ for all $j=1,2,3$. The results can be generalized to $N$-tensors with any $N\geq 3$. Both the lower and upper bounds for the tensor nuclear norm are sharp in the case $N=3$. A computable criterion for the lower bound being tight is given as well.