Temporal Hierarchical Clustering

TL;DR

This paper introduces a method for creating temporally coherent hierarchical clusterings of dynamic data by embedding point sets into ultrametric spaces and ensuring minimal distortion over time.

Contribution

It formalizes the problem of temporal hierarchical clustering, proposes an embedding-based approach, and provides bounds on the problem's approximability.

Findings

Established bounds on the approximability of the clustering problem.

Proposed a framework for embedding point sets into ultrametrics with temporal coherence.

Analyzed the trade-offs between clustering accuracy and temporal smoothness.

Abstract

We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric space, which naturally induces a hierarchical clustering of the set of points. We enforce temporal coherence among the embeddings by finding correspondences between successive pairs of ultrametric spaces which exhibit small distortion in the Gromov-Hausdorff sense. We present both upper and lower bounds on the approximability of the resulting optimization problems.