  The development of algorithms for unsupervised pattern recognition by
nonlinear clustering is a notable problem in data science. Markov clustering
(MCL) is a renowned algorithm that simulates stochastic flows on a network of
sample similarities to detect the structural organization of clusters in the
data, but it has never been generalized to deal with data nonlinearity. Minimum
Curvilinearity (MC) is a principle that approximates nonlinear sample distances
in the high-dimensional feature space by curvilinear distances, which are
computed as transversal paths over their minimum spanning tree, and then stored
in a kernel. Here we propose MC-MCL, which is the first nonlinear kernel
extension of MCL and exploits Minimum Curvilinearity to enhance the performance
of MCL in real and synthetic data with underlying nonlinear patterns. MC-MCL is
compared with baseline clustering methods, including DBSCAN, K-means and
affinity propagation. We find that Minimum Curvilinearity provides a valuable
framework to estimate nonlinear distances also when its kernel is applied in
combination with MCL. Indeed, MC-MCL overcomes classical MCL and even baseline
clustering algorithms in different nonlinear datasets.
