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.

While neural networks can solve complex geometric problems, as demonstrated by systems like AlphaGeometry, we have limited understanding of how they internally represent and reason about spatial relationships. In this work, we investigate how neural networks develop internal spatial understanding by training Graph Neural Networks and Transformers to predict point positions on a discrete 2D grid from geometric constraints that describe hidden figures. We show that both models develop interpretable internal representations that mirror the geometric structure of the problems they solve. Specifically, we observe that point embeddings self-organize into 2D grid structures during training, and during inference, the models iteratively construct the hidden geometric figures within their embedding spaces. Our analysis reveals how reasoning complexity correlates with prediction accuracy, and shows that models solve constraints through an iterative refinement process, which might resemble continuous optimization. We also find that Graph Neural Networks prove more suitable than Transformers for this type of structured constraint reasoning and scale more effectively to larger problems. These findings provide initial insights into how neural networks can develop structured understanding and contribute to their interpretability.