<p>Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)?</p>

<p>For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has mode-ranks</p>

<ul>
<li>rank-mode1 = 15 (i.e. the matrix-rank after unfolding the 3D-tensor along mode1)</li>
<li>rank-mode2 = 16 (ditto)</li>
<li>rank-mode3 = 10 (ditto)</li>
</ul>

<p>Does it make sense to normalize the ranks by their respective mode-lengths"?</p>

<ul>
<li>normalized rank-mode1 = 15/60</li>
<li>normalized rank-mode2 = 16/120</li>
<li>normalized rank-mode3 = 10/30</li>
</ul>

<p>And, for the example above, can I use this normalized rank to infer a simpler "structure" along mode2-unfolding?</p>

<p>OR does the lowest rank, regardless of mode-direction, define the "simplest" structure direction?</p>

<p>EDIT: Motivation - I'm working with reconstruction algorithms and trying to understand the significance of rank along different modes
Thank you</p>

CC BY-SA 3.0
2013-06-05T18:30:30.443Z
I think it would be helpful if you provide some background and motivation so people can better understand what you are looking for.
Kaveh
CC BY-SA 3.0
2013-06-05T21:05:56.567Z
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?
Joshua Grochow
CC BY-SA 3.0
2013-06-06T04:25:33.190Z
@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
<linear-algebra><matrices><tensor-rank>
normalized tensor rank
msg
# normalized tensor rank

(Asked by: username_0 on 2013-06-05)

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

**Tags:** linear-algebra, matrices, tensor-rank
> **Comments:**
> **username_1**: I think it would be helpful if you provide some background and motivation so people can better understand what you are looking for.  
> **username_2**: 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?  
> **username_0**: @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?  

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## 0 answers

