  We discuss topological aspects of cluster analysis and show that inferring
the topological structure of a dataset before clustering it can considerably
enhance cluster detection: theoretical arguments and empirical evidence show
that clustering embedding vectors, representing the structure of a data
manifold instead of the observed feature vectors themselves, is highly
beneficial. To demonstrate, we combine manifold learning method UMAP for
inferring the topological structure with density-based clustering method
DBSCAN. Synthetic and real data results show that this both simplifies and
improves clustering in a diverse set of low- and high-dimensional problems
including clusters of varying density and/or entangled shapes. Our approach
simplifies clustering because topological pre-processing consistently reduces
parameter sensitivity of DBSCAN. Clustering the resulting embeddings with
DBSCAN can then even outperform complex methods such as SPECTACL and
ClusterGAN. Finally, our investigation suggests that the crucial issue in
clustering does not appear to be the nominal dimension of the data or how many
irrelevant features it contains, but rather how \textit{separable} the clusters
are in the ambient observation space they are embedded in, which is usually the
(high-dimensional) Euclidean space defined by the features of the data. Our
approach is successful because we perform the cluster analysis after projecting
the data into a more suitable space that is optimized for separability, in some
sense.
