The first time we ran the tests for the Section A Second Random Example of the 2 to the 9th tensors, we used the ALS algorithm at fixed rank rather than growing the rank (see ALS Performance Notes). The fixed-rank algorithm performed much worse. In the following table we give the corresponding visualizations produced by the two algorithms.
Rank | Fixed Rank | Growing Rank |
---|---|---|
16 | ![]() |
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32 | ![]() |
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40 | ![]() |
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48 | ![]() |
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Zoomed out to the square , the fixed-rank
visualization of
gives
Since the ALS routine was called for a grid of tensors in
, the points outside this region indicate
that the fixed-rank ALS sometimes found the ‘best’ approximation to be
quite far from the target tensor in terms of its projection as well as
its angle.