Extensions to the Standard PSOM Algorithm
From the previous examples, we clearly see that
in general we have to address the problem of multiple minima,
which we combine with a solution to the problem of local minima.
This is the subject of the next section. In the following,
section 6.2 describes a way of employing the multiway mapping
capabilities of the PSOM algorithm for additional purposes, e.g.
in order to simultaneously comply to auxiliary constraints given
to resolve redundancies.
If an increase in mapping accuracy is desired,
one usually increases the number of free parameters, which
translates in the PSOM method to more training points per
parameter axis. Here we encounter two shortcomings with the
original approach:
The choice
of polynomials as basis functions of increasing order leads to
unsatisfactory convergence properties. Mappings of sharply
peaked functions can force a high degree interpolation
polynomial to strong oscillations, spreading across the entire
manifold.
Both aspects motivate two extensions to the
standard PSOM approach: the “Local-PSOMs” and the “Chebyshev-spaced
PSOM”, which are the focus of the Sec. 6.3 and 6.4.
reoccurs except for the middle region and the
right side, close to the last reference point. First, we want to
note that the problem can be detected here by monitoring the
residual distance
dist
(respectively the cost
function Es)
which is here the horizontal distance of the found completion
(close to w)
and the input x.
Second, this problem can be solved by
re-starting the search: suitable restart points are the
distance-ranked list of node locations
a
found in the first place (e.g. the
10th points probes the start locations at node 3,2,1,4). The
procedure stops, if a satisfying solution (low residual cost
function) or a maximum number of trials is reached. Fig. 6.1b
demonstrates this multi-start procedure and depicts the
correctly found solutions.
In case the task is known to involve a
consecutive sequence of query points, it is perfectly reasonable
to place the previous best-match location
s
at the head position of the list of
start locations. Furthermore, the multi-start technique is also
applicable to find multiple best-match solutions. However, extra
effort is required to find the complete list of compatible
solutions. E.g. in the middle region of the depicted example
Fig. 6.1, at least two of the three solutions will be found.