Data Analysis Using Manifold Learning: The RDSF Algorithm

In modern information systems, data analysis is pivotal for uncovering hidden patterns and extracting meaningful insights. Visualizing the behavior of real-valued functions over high-dimensional domains enables researchers to gain intuitive understanding of complex systems. Traditionally, mathematical problems are tackled analytically; however, a data-driven approach—where synthetic samples are explored systematically—can reveal new directions and solution spaces. In this paper, we introduce the RDSF algorithm, short for \textit{Reducing the Dimension of the Space of Independent and Dependent Variables of Real-Valued Functions}. This algorithm leverages manifold learning, specifically Multidimensional Scaling (MDS), to reduce the dimensionality of both input and output spaces, allowing effective visualization and analysis of function behaviors. We present a unified framework for analytical and visual exploration of mathematical and engineering problems using RDSF. The utility of RDSF is demonstrated through multiple case studies, including approximate solutions to partial differential equations (PDEs) and topological analysis of the distribution of prime numbers. These applications reveal that even abstract mathematical domains can benefit significantly from visual, data-oriented perspectives. The proposed framework is general, adaptable, and opens new avenues for exploration across disciplines.