We now study the case of tensors over the real and complex numbers. Our goals are
In principle one can never determine the maximum separation rank using a numerical routine such as this, since one could be unlucky and miss it. Generally one can only hope to determine the typical separation rank.
In [KRUSKA1989], Kruskal claims to have shown rank , but his results were not published. Jos ten Berge says that he saw this 20 page proof but discarded it when Roberto Rocci provided a 4 page proof in 1993, which is also unpublished but is available. A full proof is published in [BRE-HU2013].
We choose , , and randomly in . Since these three tensors are in , any tensor on the line connecting any two of them is in , and a tensor in the plane is generically in , which we suppose is large enough to capture general tensors.
We first produce a visualization of on , and obtain
[data file 3x3x3pr4random.dat] There are only a few scattered black points. Thus we conclude that the maximum separation rank of a tensor over the real numbers is more than 4.
Next, using the same set of random , , and , we produce a visualization of , and obtain
[data file 3x3x3pr5random.dat] Black points are distributed over the whole plane, with no apparent voids. We interpret this as validating the result that the maximum rank is 5.
Finally, using the same set of random , , and , we first produce a visualization for , expecting it to be solid black. Since the line connecting any two is in , these lines appear in the visualization, and help to locate the . We obtain the visualization
[data file 3x3x3pr6random.dat] The occasional lighter point is attributed to failure in the routine that tries to fit a tensor on the plane with a tensor in .
We now consider a specific example that was suggested as possibly interesting. The target tensor, which has nominal rank 5, is
which can also be viewed as
Using the plane defined by the three tensors
the point is in their center.
We first produce a visualization of on , and obtain
[data file 3x3x3pr3X5.dat] Since the region in the vicinity of has no black points, we conclude it is not rank 3. We next produce a visualization of and obtain
[data file 3x3x3pr4X5.dat] A large number of test points were needed to produce this visualization. We used more points near the center to try to get better resolution there. Although there are not that many black points, they are spread throughout the visualization with no apparent voids. We also tried zooming in on a relatively light region near the center and adding many more points there, and were then able to find dark points. Based on this behavior, we conclude that is probably of rank 5 but reachable as a limit of rank 4 tensors.
The rank over the complex numbers is at most the rank over the real numbers, so we now investigate whether or not the maximum rank is strictly smaller for tensors.
We choose and randomly in , so that a tensor in the plane is generically in , which we know is large enough to capture general tensors. We then produce a visualization of and obtain
[data file 3x3x3pr4randomC.dat] Since there are few black points, we conclude the maximum rank over the complex numbers is more than 4, and hence is 5. As an additional check, we use the same and and produce a visualization of , and obtain
[data file 3x3x3pr5randomC.dat]
We now reconsider the tensor , which we studied in the real case. We use the complex line defined by
We first produce a visualization of on , and obtain
[data file 3x3x3pr3X5C.dat] Since there are essentially no black points, we then produce a visualization of and obtain
[data file 3x3x3pr4X5C.dat] This visualization required quite a lot of points and effort. The scattered dark blue points indicate that rank 4 points are present (or nearby) but that they are hard to construct. We zoomed in to and added points near , and obtain
Based on this behavior, we again conclude that is probably of rank 5 but reachable as a limit of rank 4 tensors.