Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)?
For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has mode-ranks
- rank-mode1 = 15 (i.e. the matrix-rank after unfolding the 3D-tensor along mode1)
- rank-mode2 = 16 (ditto)
- rank-mode3 = 10 (ditto)
Does it make sense to normalize the ranks by their respective mode-lengths"?
- normalized rank-mode1 = 15/60
- normalized rank-mode2 = 16/120
- normalized rank-mode3 = 10/30
And, for the example above, can I use this normalized rank to infer a simpler "structure" along mode2-unfolding?
OR does the lowest rank, regardless of mode-direction, define the "simplest" structure direction?
EDIT: Motivation - I'm working with reconstruction algorithms and trying to understand the significance of rank along different modes Thank you