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A Uniqueness Theorem for Clustering
Machine Learning · Submitted 9 May 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main result of this paper is a set of abstract properties that characterize the Single-Linkage clustering function.
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The abstract, sentence by sentence
Despite the widespread use of Clustering, there is distressingly little general theory of clustering available. Questions like "What distinguishes a clustering of data from other data partitioning?", "Are there any principles governing all clustering paradigms?", "How should a user choose an appropriate clustering algorithm for a particular task?", etc. are almost completely unanswered by the existing body of clustering literature. We consider an axiomatic approach to the theory of Clustering. We adopt the framework of Kleinberg, [Kle03]. By relaxing one of Kleinberg's clustering axioms, we sidestep his impossibility result and arrive at a consistent set of axioms. We suggest to extend these axioms, aiming to provide an axiomatic taxonomy of clustering paradigms. Such a taxonomy should provide users some guidance concerning the choice of the appropriate clustering paradigm for a given task. The main result of this paper is a set of abstract properties that characterize the Single-Linkage clustering function. This characterization result provides new insight into the properties of desired data groupings that make Single-Linkage the appropriate choice. We conclude by considering a taxonomy of clustering functions based on abstract properties that each satisfies.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Despite the widespread use of Clustering, there is distressingly little general theory of clustering available. | gap 0.69 | 0.04 | 0.01 | asserted 1.00 |
| s1 | Questions like "What distinguishes a clustering of data from other data partitioning?", "Are there any principles governing all clustering paradigms?", "How should a user choose an appropriate clustering algorithm for a particular task?", etc. are almost completely unanswered by the existing body of clustering literature. | gap 0.99 | 0.03 | 0.02 | asserted 1.00 |
| s2 | We consider an axiomatic approach to the theory of Clustering. | contribution 0.59 | 0.89 | 0.02 | asserted 0.95 |
| s3 | We adopt the framework of Kleinberg, [Kle03]. | method 0.73 | 0.73 | 0.02 | asserted 0.98 |
| s4 | By relaxing one of Kleinberg's clustering axioms, we sidestep his impossibility result and arrive at a consistent set of axioms. | method 0.47 | 0.93 | 0.06 | asserted 1.00 |
| s5 | We suggest to extend these axioms, aiming to provide an axiomatic taxonomy of clustering paradigms. | contribution 0.55 | 0.69 | 0.04 | hedged possibility 0.88 |
| s6 | Such a taxonomy should provide users some guidance concerning the choice of the appropriate clustering paradigm for a given task. | interpretation 0.86 | 0.17 | 0.04 | hedged possibility 0.86 |
| s7 | The main result of this paper is a set of abstract properties that characterize the Single-Linkage clustering function. | result 0.95 | 0.94 | 0.96 | asserted 1.00 |
| s8 | This characterization result provides new insight into the properties of desired data groupings that make Single-Linkage the appropriate choice. | interpretation 1.00 | 0.86 | 0.11 | asserted 0.94 |
| s9 | We conclude by considering a taxonomy of clustering functions based on abstract properties that each satisfies. | contribution 0.58 | 0.86 | 0.16 | asserted 0.75 |
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Original abstract
Despite the widespread use of Clustering, there is distressingly little general theory of clustering available. Questions like "What distinguishes a clustering of data from other data partitioning?", "Are there any principles governing all clustering paradigms?", "How should a user choose an appropriate clustering algorithm for a particular task?", etc. are almost completely unanswered by the existing body of clustering literature. We consider an axiomatic approach to the theory of Clustering. We adopt the framework of Kleinberg, [Kle03]. By relaxing one of Kleinberg's clustering axioms, we sidestep his impossibility result and arrive at a consistent set of axioms. We suggest to extend these axioms, aiming to provide an axiomatic taxonomy of clustering paradigms. Such a taxonomy should provide users some guidance concerning the choice of the appropriate clustering paradigm for a given task. The main result of this paper is a set of abstract properties that characterize the Single-Linkage clustering function. This characterization result provides new insight into the properties of desired data groupings that make Single-Linkage the appropriate choice. We conclude by considering a taxonomy of clustering functions based on abstract properties that each satisfies.
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