N

= {ye Nfc: yt e {0,1}Vi} = {«1, «2, •••, «c}.

In Eq. (1) 0 is the zero vector in • Nhc is the canonical (unit vector) basis of Euclidean c-space, so et- = (0, •••, 1 , •••, 0) , the ith vertex of Nhc, is the crisp label for class i,t 1 < i < c. Nj-C contains the fuzzy (and probabilistic) label vectors; Npc contains possibilistic label vectors. Note that Nhc c Nfc c Npc. The three kinds of noncrisp labels are collectively called soft labels. Figure 4 is a sketch of the structure embodied by Eqs. (1)-(3) for c = 3.

A c-partition of X is a c xn matrix U = [uik ]. There are three sets of nondegenerate c-partitions, whose columns {Uk} correspond to the three types of label vectors:

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