HyperANFIS: Enhancing Rule Representation and Interpretability in Adaptive Neuro-Fuzzy Systems via Hyperbolic Geometry

Machine learning models achieve high predictive accuracy in regression tasks, but their deployment in safety-critical and regulated domains requires interpretability. While fuzzy rule-based systems offer transparent, linguistically explicit interpretable models, Mamdani-style fuzzy regression remains underrepresented in modern machine learning software libraries. This paper presents an interpretable regression extension for the Ex-Fuzzy library, enabling Mamdani fuzzy inference with scalar consequents learned directly from data. For this, a target-aware partition initialisation strategy based on Fuzzy C-Means clustering is introduced, in which linguistic variables are derived from an augmented input-output space to emphasise output-relevant regions of the feature space. The proposed extension is evaluated on ten regression datasets from the KEEL repository, comparing Gaussian and trapezoidal partition strategies against standard baselines including linear regression, multilayer perceptron, and random forests. Experimental results show that Gaussian partitions consistently outperform uniform trapezoidal partitions, achieving a mean coefficient of determination of approximately 0.86 while producing compact rule bases of 10-15 human-readable rules. The proposed implementation provides a transparent and competitive alternative to black-box regression models, supporting practical interpretability with competitive predictive performance.

Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce \textbf{HY}perbolic \textbf{D}ynamic \textbf{R}epresentation \textbf{A}rchitecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincaré ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling efficient computation and exact symbolic rule extraction. As a result, each node (and the entire model) can be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine-McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.