For a 3-tensor of dimensions <sup>I1</sup>×<sup>I2</sup>×<sup>I3</sup>, 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 equation presented times the nuclear norm of the matrix flattening in mode j, for all j=1,2,3. The results can be generalized with some modifications to N-tensors with N>3. Both the lower and the upper bounds for the tensor nuclear norm are sharp in the case N=3. A computable sufficient criterion for the lower bound being tight is given as well.
Hu, S. (2015). Relations of the nuclear norm of a tensor and its matrix flattenings. Linear Algebra and Its Applications, 478, 188–199. https://doi.org/10.1016/j.laa.2015.04.003