The first two steps in the optimization paradigm of Section 3.8 are constructing the mathematical model and defining a class of operators. These steps are mutual in the sense that one cannot begin to analyze a class of operators without knowing the mathematical structure of the operator domain, which in the case of random phenomena means specification of the underlying random process. In the case of clustering, given a finite point set, a cluster operator partitions the set, the input being the point set and the output being a partition, which is a disjoint collection of subsets whose union is the full point set. Just as a specific signal is a realization of a random function modeling a signal-generation process, a specific point set is a realization of a random point set modeling a point-set-generation process. The study of clustering must begin with a random set, or else it is scientifically meaningless. Moreover, the random set must be sufficiently endowed that it supports the analysis of partitioning, including partition error. Optimization involves choosing a cluster operator that minimizes the partition error.
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