Relations of the nuclear norm of a tensor and its matrix flattenings

For a 3-tensor of dimensions I1×I2×I3I1×I2×I3, 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 min⁡{Ii:i≠j} times the nuclear norm of the matrix flattening in mode j  , for all j=1,2,3j=1,2,3. The results can be generalized with some modifications to N  -tensors with N>3N>3. Both the lower and the upper bounds for the tensor nuclear norm are sharp in the case N=3N=3. A computable sufficient criterion for the lower bound being tight is given as well.