Clustering is a widely used technique, with applications ranging from data mining, bioinformatics and image analysis to marketing, psychology, and city planning. Despite the practical importance of clustering, there is very limited theoretical analysis of the topic. We make a step towards building theoretical foundations for clustering by carrying out an abstract analysis of two central concepts in clustering; clusterability and clustering quality. We compare a number of notions of clusterability found in the literature. While all these notions attempt to measure the same property, and all appear to be reasonable, we show that they are pairwise inconsistent. In addition, we give the first computational complexity analysis of a few notions of clusterability. In the second part of the thesis, we discuss how the quality of a given clustering can be defined (and measured). Users often need to compare the quality of clusterings obtained by different methods. Perhaps more importantly, users need to determine whether a given clustering is sufficiently good for being used in further data mining analysis. We analyze what a measure of clustering quality should look like. We do that by introducing a set of requirements (`axioms') of clustering quality measures. We propose a number of clustering quality measures that satisfy these requirements.

Clustering is a widely used technique, with applications ranging from data mining, bioinformatics and image analysis to marketing, psychology, and city planning. Despite the practical importance of clustering, there is very limited theoretical analysis of the topic. We make a step towards building theoretical foundations for clustering by carrying out an abstract analysis of two central concepts in clustering; clusterability and clustering quality. We compare a number of notions of clusterability found in the literature. While all these notions attempt to measure the same property, and all appear to be reasonable, we show that they are pairwise inconsistent. In addition, we give the first computational complexity analysis of a few notions of clusterability. In the second part of the thesis, we discuss how the quality of a given clustering can be defined (and measured). Users often need to compare the quality of clusterings obtained by different methods. Perhaps more importantly, users need to determine whether a given clustering is sufficiently good for being used in further data mining analysis. We analyze what a measure of clustering quality should look like. We do that by introducing a set of requirements (`axioms') of clustering quality measures. We propose a number of clustering quality measures that satisfy these requirements.

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