A note on ALS failureΒΆ

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 _images/2to9pr16random2fixed.jpg _images/2to9pr16random2.jpg
32 _images/2to9pr32random2fixed.jpg _images/2to9pr32random2.jpg
40 _images/2to9pr40random2fixed.jpg _images/2to9pr40random2.jpg
48 _images/2to9pr48random2fixed.jpg _images/2to9pr48random2.jpg

Zoomed out to the square [-2,3]\times[-2,3], the fixed-rank visualization of S_{48} gives

_images/2to9pr48random2out.jpg

Since the ALS routine was called for a grid of tensors in [-1,2]\times[-1,2], 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.

Previous topic

2 to the 9th Tensors

Next topic

Software

This Page