Similarity and Pattern Recognition

This paper formally defines similarities as tolerance relations, which are reflexive and symmetric binary relations. An abstract set with a similarity is called a tolerance space. The training data set in a learning task is a given database of independent identically distributed random pairs (Xi, Yi), where each Xi is a record and Yi is its label: Yi ∈ {0, 1}. The goal of the learning is to design a classifier of which the error probability is near to the theoretical limitation, the Bayes error. The learning process consists of finding a similarity of feature vectors ψ(Xi)’s and the learning result is a representative data clustering on the tolerance space of feature vectors. The information about a record X derived from the representative clustering is the set of representatives similar to the feature vector ψ(X). The percentage of the records of class 1 in the intersection of these representative clusters is used to estimate the conditional probability of Y = 1. This paper defines a θ-classifier, which assigns the record to class 1 if the conditional probability is larger than the threshold θ. If the clustering is a partition, the threshold θ = 1 2 minimizes error probability in the training data set. In general, an optimal θ-classifier has a different threshold. The experiments show the trade-off between the number of clusters and the error probabilities of the optimal