dimensionality reduction, F-transform, eigenvectors

In this contribution, we focus on a geometrically interpretable way how to initiate the dimensionality reduction process utilizing the F-transform on a space with a fuzzy partition. We consider a pre-clustered initial data and propose a construction of the graph Laplacian that represents a space with a fuzzy partition. Our basic idea is to connect each point in the k-th cluster, endowed with k-th basic function, by a graph edge with the weight determined by that basic function and then, after certain transformation, obtain graph Laplacian. The spectral analysis of the corresponding Laplacian operators shows the existence of generalized eigenvectors of three types: constant vector, vector that is constant except for one coordinate and vectors that belong to a certain eigenspace. These vectors are compared with the discrete higher degree F-transform components. In the similar manner, we would like to characterize eigenvectors of other commonly used Laplacian operators known from the literature.