Gaussian model for closed curves

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.