Fuzzy partition, Closeness, F-transform, Preimage problem, SVD

Firstly, we consider a discrete universe X with a fuzzy partition in which we distinguish between points and nodes. Fuzzy partition is a natural way how to establish closeness within X. The binary relation of closeness is defined on pairs of the type (node, point) –⁠ in such a way that its values are equal to the corresponding basic function values. This gives rise to a rectangular adjacency matrix W describing a certain graph structure G on the data. The vertices of G are nodes and points, and its edges connect only those pairs with positive closeness. By this initial setting, we introduce a space that is more general than a metric space. Next, we consider the set F of all real functions defined on the graph vertices and the set H of all real functions defined on the graph nodes. The F-transform linearly maps F to H. Therefore, the direct F-transform of a function u in F is the image of D^{-1}W, where D is the weighted diagonal matrix of W, i.e. F[u]=D^{-1}Wu. The goal of this contribution is to characterize all preimages of F[u], given u. We show that the set of preimages of F[u] is a similarity (equivalence) class of u, where the similarity is established on F in the sense that similar functions are mapped on the same node function in H. Moreover, this approach provides a means how to ``reconstruct'' a representative function from the similarity class, given its F-transform components. Finally, the solution to the above discussed preimage problem is presented from three different perspectives, utilizing a singular value decomposition of W. The aforementioned propositions are supported by numerical experiments.