# normalized tensor rank

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

• I think it would be helpful if you provide some background and motivation so people can better understand what you are looking for. Jun 5 '13 at 18:30
• What do you mean by "simplest"? Also, when you say "structure direction" (a term I am unfamiliar with but can kind of guess what you mean), do you only allow the three natural choices, or do you allow "linear combinations" of those directions? Jun 5 '13 at 21:05
• @joshua By simplest I mean the mode that has the smallest rank (i.e the smallest number of linearly independent vectors). Not sure what you mean by lc of the directions. For example, if a particular mode unfolding has rank 10 with full rank being 100, is that any "better" than a different mode unfolding (same tensor) with rank 15 out of a full rank 20?
– msg
Jun 6 '13 at 4:25