Natural Methods of Unsupervised Topological Alignment

In the paper, we present a comparison analysis of the methods of the topological alignment and extract the main mathematical principles forming the base of the concept. The main narrative is devoted to the so-called coupled methods dealing with the data sets of various natures. As a main theoretical result, we obtain harmonious generalizations of the graph Laplacian and kernel-based methods with the central idea to find a natural structure coupling data sets of various natures. Finally, we discuss prospective applications and consider far-reaching generalizations related to the hypercomplex numbers.