In continuous RL, the labeling state for each site is represented as a
position vector; for i, it is
subject to the following feasibility constraints
This was interpreted as a fuzzy assignment model [Rosenfeld et al. 1976]. The real value
reflects the strength with which i is assigned label
I. Each 
lies in a hyperplane in the non-negative quadrant
portion (simplex) of the multi-dimensional real space 
as
shown in the shaded area of Fig.8.3. The feasible
labeling assignment space is
The set 
is called a labeling
assignment. The feasible space for labeling
assignments is the following product
The final solution 
must be unambiguous
with 
meaning that i is unambiguously labeled I.
This gives the unambiguous spaces for 
as
So every 
in 
is one of the vectors 
,
and 
corresponding to the ``corners''
of the simplex 
( cf.
Fig.8.3). The
unambiguous labeling assignment space is
which consists of the corners of the space 
. Non-corner
points in the continuous space 
provide routes to the corners of
, i.e.
the points in the discrete space 
. An unambiguous
labeling assignment is related to the corresponding discrete labeling by
The unambiguity may be satisfied automatically upon convergence of some iterative algorithms or if not, has to be enforced e.g. using a maximum selection (winner-take-all) operation. However, the forced unambiguous solution is not necessarily a local minimum. How to ensure the result of a continuous RL algorithm to be an unambiguous one has been largely ignored in the literature.
Figure 8.3: The labeling space for 
with three labels. The feasible
space is shown as the shaded area (simplex) and the unambiguous space
consists of the three corner points.
