A Fractional Probabilistic Self-Organizing Map (FRAC-PRSOM) via Caputo-Fabrizio Derivative for Improved Clustering Performance

Probabilistic Self-Organizing Maps (PRSOM) are effective for visualizing complex patterns in large-scale data due to their neural network structure and probabilistic reasoning. However, conventional PRSOM relies on ordinary derivatives, limiting its ability to uncover latent structures and capture long-range dependencies. This study presents a novel Fractional Probabilistic Self-Organizing Map (FRAC-PRSOM) that integrates the Caputo-Fabrizio (CF) fractional derivative into the learning process. Unlike conventional integer-order methods, the non-singular exponential kernel of the Caputo-Fabrizio derivative enables smoother parameter evolution, improved numerical stability, and enhanced representation of complex cluster structures while ensuring smooth recovery of the classical PRSOM formulation as α→1−. The proposed model is validated on five benchmark datasets using six complementary metrics, including geometry-based and model-based criteria. Results consistently show that FRAC-PRSOM reduces final log-likelihood error by an average of 86.5% across all datasets and achieves the lowest average ranks in clustering performance, establishing fractional-order learning as a highly effective paradigm for probabilistic self-organizing maps. Statistical significance is confirmed via Friedman and Holm tests. Additionally, an image compression task further demonstrates the practical effectiveness of the proposed model, the FRAC-PRSOM achieves superior peak signal-to-noise ratio (PSNR) values (up to 83.59 dB) compared to PRSOM and the Enhanced Fractional PRSOM (EF-PRSOM).