Abstract
A new fuzzy c-means algorithms for data with tolerance is proposed by
introducing a penalty term in feature space.
Its idea is derived from that support vector machine
introducing a penalty term for "soft margin" in feature space.
In the proposed method,
the data is allowed to move for minimizing the corresponding objective function
but this move-ness is controlled by the penalty term.
First, an optimization problem is shown
by introducing tolerance with conventional fuzzy c-means algorithm in feature space.
Second, Karush-Kuhn-Tucker~(KKT) conditions of the optimization problem is considered.
Third, an iterative algorithm is proposed by re-expressing the KKT conditions
using kernel trick.
Fourth, another iterative algorithm is proposed for fuzzy classification function,
which shows how prototypical an arbitrary point in the data space is to the obtained each cluster by extending the membership to the whole spa
ce.
Last, some numerical examples are shown.
