  Gaussian Mixture Models (GMM) do not adapt well to curved and strongly
nonlinear data. However, we can use Gaussians in the curvilinear coordinate
systems to solve this problem. Moreover, such a solution allows for the
adaptation of clusters to the complicated shapes defined by the family of
functions. But still, it is challenging to model clusters as closed curves
(e.g., circles, ellipses, etc.). In this work, we propose a density
representation of the closed curve, which can be used to detect the complicated
templates in the data. For this purpose, we define a new probability
distribution to model closed curves. Then we construct a mixture of such
distributions and show that it can be effectively trained in the case of the
one-dimensional closed curves.
