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 | ||
32 | ||
40 | ||
48 |
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.