  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.
