On the Discrepancy Between Kleinberg's Clustering Axioms and $k$-Means Clustering Algorithm Behavior

宽间隙与聚类公理

Wide Gaps and Clustering Axioms

针对k-means违反Kleinberg一致性公理的问题，本文提出变分与残差k-可分性两种可聚类性条件，证明在此条件下k-means满足一致性，并通过算法扩展近似满足丰富性公理，从而调和二者关系。

Possibility results for graph clustering: A novel consistency axiom

Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg's Consistency axiom, and the impossibility result. Namely, we describe a clustering algorithm, Morse Clustering, inspired by Morse Theory in Differential Topology, which satisfies Kleinberg's original axioms with Consistency replaced by Monotonic Consistency. Morse clustering uncovers the underlying flow structure on a set or…

High-Dimensional Wide Gap $k$-Means Versus Clustering Axioms

Kleinberg's axioms for distance based clustering proved to be contradictory. Various efforts have been made to overcome this problem. Here we make an attempt to handle the issue by embedding in high-dimensional space and granting wide gaps between clusters.

A Uniqueness Theorem for Clustering

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 si…

A Clustering Preserving Transformation for k-Means Algorithm Output

This note introduces a novel clustering preserving transformation of cluster sets obtained from $k$-means algorithm. This transformation may be used to generate new labeled data{}sets from existent ones. It is more flexible that Kleinberg axiom based consistency transformation because data points in a cluster can be moved away and datapoints between clusters may come closer together.

层次聚类可以同时满足丰富性、一致性和尺度不变性

Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance

本文证明层次聚类可同时满足尺度不变性、丰富性和一致性公理，构造了多种可容许方法，并揭示了其偏序结构中的多样性与共同主干。