Fuzzy Relational Models: Convolution Techniques and Optimization

Convolutional operations have been one of the mechanisms of functional processing of neural networks, especially present as a part of convolutional neural networks. Fuzzy convolution (composition operation) has been widely explored. within the framework of fuzzy relational equations, its integration into computational architectures such as neural networks remains underexplored. This study introduces a framework that formally extends conventional convolution into the domain of fuzzy set theory through the development of fuzzy convolution operations. We formulate and solve an optimization problem aimed at fine-tuning fuzzy convolution kernels using established fuzzy relational structures, thereby enhancing the interpretability of neural processing. Several t-norms and t-conorms that implement convolution operators are examined within the framework of s-t and t-s convolutions (compositions) of fuzzy relations. A detailed derivation of the optimization schemes is presented. Several experiments on images are conducted, demonstrating that even with a data size reduction of up to 75%, the method can still effectively reconstruct images by optimizing the parameters of the relational architecture.