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

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.