closeness; closeness matrix; closeness space; function similarity; fuzzy partition; fuzzy transform; preimage problem; singular value decomposition

In this article, we focus on discrete data processing. We propose to use the concept of closeness, which is less restrictive than a metric, to describe a certain relationship between objects. We establish a fuzzy partition of a given set of objects in a way that admits a closeness space to emerge. The fuzzy (F-) transform is a tool that maps objects with common characteristics to the same discrete image—the direct F-transform. We are interested in the inverse (preimage) problem: How can we describe the class of all functions mapped onto the same direct F-transform? In this manuscript, we focus on this preimage problem, formulated accordingly. Its solution is presented from three different points of view and shows which functions belong to the same class determined by a given image (by the direct F-transform). Conditions under which a solution to the preimage problem is given by the inverse F-transform over the same fuzzy partition, or by transforming a given image using a new system of basic functions, are formulated. The developed theory contributes to a better understanding of ill-posed problems that are typical for machine learning. The appendix contains illustrative numerical examples.