Representation learning has emerged as a crucial focus in machine and deep learning, involving the extraction of meaningful and useful features and patterns from the input data, thereby enhancing the performance of various downstream tasks such as classification, clustering, and prediction. Current mainstream representation learning methods primarily rely on non-linear data mining techniques such as kernel methods and deep neural networks to extract abstract knowledge from complex datasets. However, most of these methods are black-box, lacking transparency and interpretability in the learning process, which constrains their practical utility. To this end, this paper introduces a novel representation learning method grounded in an interpretable fuzzy rule-based model. Specifically, it is built upon the Takagi-Sugeno-Kang fuzzy system (TSK-FS) to initially map input data to a high-dimensional fuzzy feature space through the antecedent part of the TSK-FS. Subsequently, a novel differentiable optimization method is proposed for the consequence part learning which can preserve the model's interpretability and transparency while further exploring the nonlinear relationships within the data. This optimization method retains the essence of traditional optimization, with certain parts of the process parameterized corresponding differentiable modules constructed, and a deep optimization process implemented. Consequently, this method not only enhances the model's performance but also ensures its interpretability. Moreover, a second-order geometry preservation method is introduced to further improve the robustness of the proposed method. Extensive experiments conducted on various benchmark datasets validate the superiority of the proposed method, highlighting its potential for advancing representation learning methodologies.

Generative Models (GMs), particularly Large Language Models (LLMs), have garnered significant attention in machine learning and artificial intelligence for their ability to generate new data by learning the statistical properties of training data and creating data that resemble the original. This capability offers a wide range of applications across various domains. However, the complex structures and numerous model parameters of GMs make the input-output processes opaque, complicating the understanding and control of outputs. Moreover, the purely data-driven learning mechanism limits GM's ability to acquire broader knowledge. There remains substantial potential for enhancing the robustness and generalization capabilities of GMs. In this work, we introduce the fuzzy system, a classical modeling method that combines data and knowledge-driven mechanisms, to generative tasks. We propose a novel Generative Fuzzy System framework, named GenFS, which integrates the deep learning capabilities of GM with the interpretability and dual-driven mechanisms of fuzzy systems. Specifically, we propose an end-to-end GenFS-based model for sequence generation, called FuzzyS2S. A series of experimental studies were conducted on 12 datasets, covering three distinct categories of generative tasks: machine translation, code generation, and summary generation. The results demonstrate that FuzzyS2S outperforms the Transformer in terms of accuracy and fluency. Furthermore, it exhibits better performance on some datasets compared to state-of-the-art models T5 and CodeT5.