Coarse-Refinement Dilemma: On Generalization Bounds for Data Clustering

TL;DR

This paper introduces a topological framework using multidimensional persistent homology to analyze the generalization capabilities of clustering algorithms, addressing the coarse-refinement dilemma and proposing new theoretical insights.

Contribution

It presents a novel topological approach to define and analyze the data clustering problem, establishing a new generalization bound and clarifying the coarse-refinement dilemma.

Findings

Multidimensional persistent homology effectively measures divergence among clustering models.

The coarse-refinement dilemma parallels the bias-variance trade-off in clustering.

A consistency criterion for clustering models is proposed based on topological analysis.

Abstract

The Data Clustering (DC) problem is of central importance for the area of Machine Learning (ML), given its usefulness to represent data structural similarities from input spaces. Differently from Supervised Machine Learning (SML), which relies on the theoretical frameworks of the Statistical Learning Theory (SLT) and the Algorithm Stability (AS), DC has scarce literature on general-purpose learning guarantees, affecting conclusive remarks on how those algorithms should be designed as well as on the validity of their results. In this context, this manuscript introduces a new concept, based on multidimensional persistent homology, to analyze the conditions on which a clustering model is capable of generalizing data. As a first step, we propose a more general definition of DC problem by relying on Topological Spaces, instead of metric ones as typically approached in the literature. From that, we show that the DC problem presents an analogous dilemma to the Bias-Variance one, which is here referred to as the Coarse-Refinement (CR) dilemma. CR is intended to clarify the contrast between: (i) highly-refined partitions and the clustering instability (overfitting); and (ii) over-coarse partitions and the lack of representativeness (underfitting); consequently, the CR dilemma suggests the need of a relaxation of Kleinberg's richness axiom. Experimental results were used to illustrate that multidimensional persistent homology support the measurement of divergences among DC models, leading to a consistency criterion.