K-sets+: a Linear-time Clustering Algorithm for Data Points with a Sparse Similarity Measure

TL;DR

This paper introduces K-sets+, a linear-time clustering algorithm for data with sparse similarity measures, extending previous methods to semi-metric and similarity-only spaces, with proven convergence and practical effectiveness.

Contribution

The paper presents the K-sets+ algorithm, extending clustering to semi-metric and similarity spaces with linear complexity for sparse data, and provides convergence guarantees.

Findings

Converges in finite iterations.

Effective on synthetic and real datasets.

Linear complexity for sparse similarity matrices.

Abstract

In this paper, we first propose a new iterative algorithm, called the K-sets+ algorithm for clustering data points in a semi-metric space, where the distance measure does not necessarily satisfy the triangular inequality. We show that the K-sets+ algorithm converges in a finite number of iterations and it retains the same performance guarantee as the K-sets algorithm for clustering data points in a metric space. We then extend the applicability of the K-sets+ algorithm from data points in a semi-metric space to data points that only have a symmetric similarity measure. Such an extension leads to great reduction of computational complexity. In particular, for an n * n similarity matrix with m nonzero elements in the matrix, the computational complexity of the K-sets+ algorithm is O((Kn + m)I), where I is the number of iterations. The memory complexity to achieve that computational complexity is O(Kn + m). As such, both the computational complexity and the memory complexity are linear in n when the n * n similarity matrix is sparse, i.e., m = O(n). We also conduct various experiments to show the effectiveness of the K-sets+ algorithm by using a synthetic dataset from the stochastic block model and a real network from the WonderNetwork website.