Page 75 - Rapid Learning in Robotics

4.6 Summary 61

For training data sets with the known topology of a multi-dimensional
Cartesian grid the map manifold can be constructed directly. The re-
sulting PSOM is immediately usable — without any need for time
consuming adaptation sequences. This feature is extremely advan-
tageous in all cases where the training data can be sampled actively.
As shown later in Capt. 8, in robotics many sensorimotor transfor-
mations can be sampled in a structured manner, without any extra
cost.

Independently of how the the initial structure was found, further on-
line fine-tuning is facilitated by an on-line error minimization proce-
dure. This adaption is required for example, in the case of coarsely
sampled or noise-corrupted data, as well as in cases, where the learn-
ing system has to adapt to drifting, or suddenly changing mapping
tasks.

The multi-way mapping ability of the PSOM is facilitated by the non-
linear auto-associative completion of partial inputs. The components of
the embedding space are selected as inputs by the diagonal elements
p k of the projection matrix P. Furthermore, the coefficients p k weight
the components relative to each other. This choice can be changed
on demand.

The PSOM completion process performs an iterative search for the

best-matching parameter location s in the mapping manifold (Eq. 4.7).
This is the price for rapid learning. Fortunately, it can be kept small
by applying an adaptive second order minimization procedure (Sect. 4.5).
In conjunction with an algorithmic formulation optimized for effi-
cient computation and also for high-dimensional problems, the com-
pletion procedure converges already in a couple of iterations.

The following chapter will demonstrate the PSOM properties in a number
of examples. Later, several application examples will be presented.