Untangling Gaussian Mixtures
Eva Fluck Sandra Kiefer Christoph Standke
Tangles were originally introduced as a concept to formalize regions of high
connectivity in graphs. In recent years, they have also been discovered as a
link between structural graph theory and data science: when interpreting
similarity in data sets as connectivity between points, finding clusters in the
data essentially amounts to finding tangles in the underlying graphs. This
paper further explores the potential of tangles in data sets as a means for a
formal study of clusters. Real-world data often follow a normal distribution.
Accounting for this, we develop a quantitative theory of tangles in data sets
drawn from Gaussian mixtures. To this end, we equip the data with a graph
structure that models similarity between the points and allows us to apply
tangle theory to the data. We provide explicit conditions under which tangles
associated with the marginal Gaussian distributions exist asymptotically almost
surely. This can be considered as a sufficient formal criterion for the
separabability of clusters in the data.