The cluster structure function

TL;DR

This paper introduces the cluster structure function, a method based on algorithmic information theory to identify optimal data partitions by measuring how well each part models the data, demonstrated on real datasets.

Contribution

It proposes a novel clustering approach using the cluster structure function derived from Kolmogorov complexity, with practical approximations and real data examples.

Findings

The method effectively identifies meaningful data partitions.

It applies to datasets like MNIST and cell segmentation.

The approach provides a new theoretical foundation for clustering.

Abstract

For each partition of a data set into a given number of parts there is a partition such that every part is as much as possible a good model (an "algorithmic sufficient statistic") for the data in that part. Since this can be done for every number between one and the number of data, the result is a function, the cluster structure function. It maps the number of parts of a partition to values related to the deficiencies of being good models by the parts. Such a function starts with a value at least zero for no partition of the data set and descents to zero for the partition of the data set into singleton parts. The optimal clustering is the one chosen to minimize the cluster structure function. The theory behind the method is expressed in algorithmic information theory (Kolmogorov complexity). In practice the Kolmogorov complexities involved are approximated by a concrete compressor. We give examples using real data sets: the MNIST handwritten digits and the segmentation of real cells as used in stem cell research.