.. _ALSfailed-section-label:

A note on ALS failure
=====================

The first time we ran the tests for the Section
:ref:`2to9SecondExample-section-label` of the 2 to the 9th tensors, we
used the ALS algorithm at fixed rank rather than growing the rank (see
:ref:`ALSperform-section-label`). 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   .. image:: 2to9pr16random2fixed.*   .. image:: 2to9pr16random2.*
---- --------------------------------- --------------------------------
32   .. image:: 2to9pr32random2fixed.*   .. image:: 2to9pr32random2.*
---- --------------------------------- --------------------------------
40   .. image:: 2to9pr40random2fixed.*   .. image:: 2to9pr40random2.*
---- --------------------------------- --------------------------------
48   .. image:: 2to9pr48random2fixed.*   .. image:: 2to9pr48random2.*
==== ================================= ================================

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

.. image:: 2to9pr48random2out.*

Since the ALS routine was called for a grid of tensors in
:math:`[-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.