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

HyperANFIS is a hyperbolic extension of the adaptive neuro-fuzzy inference system (ANFIS) proposed to address the limitations of conventional ANFIS models, which generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. The paper states that existing ANFIS models generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. HyperANFIS preserves the fuzzy semantics and core architecture of conventional ANFIS while performing rule-prototype learning, rule activation, and consequent aggregation in hyperbolic space and retains the ability to generate interpretable IF-THEN rules.

The motivation for this work is that the increasingly numerous feature representations become crowded and constrained in Euclidean space, which can affect model performance and rule-generation capacity, particularly when the data or decision regions exhibit hierarchical, tree-like, or highly non-uniform structures. The paper notes that hyperbolic representation learning provides a theoretically motivated geometric alternative for modeling hierarchical, tree-like, and highly non-uniform structure because owing to its constant negative curvature, the volume of a metric ball in hyperbolic space grows exponentially with its radius, which allows a low-dimensional embedding to allocate increasing representational capacity to progressively branching hierarchies.

The contributions of the paper are threefold: We propose HyperANFIS, a model that learns global IF-THEN rules on a hyperbolic manifold and makes accurate predictions based on these rules; HyperANFIS unifies rule matching, local consequent construction, and consequent aggregation on a hyperbolic manifold, offering a new perspective on mitigating the limitations of Euclidean geometry in ANFIS; and Experiments on multiple real-world datasets show that HyperANFIS outperforms the compared ANFIS models and produces more credible IF-THEN rules.

The HyperANFIS architecture is organized into five functional layers: "The first layer maps the input and all rule centers from a shared origin tangent space to the selected hyperbolic representation, and then evaluates their intrinsic geodesic distances. The second layer converts each sample-to-rule distance into one scalar radial membership. The third layer normalizes the log-memberships across rules to obtain firing strengths. The fourth layer constructs a rule-specific consequent from the intrinsic local coordinates of the sample. The fifth layer maps these consequents to the output manifold, aggregates them through a firing-weighted Fréchet mean, and decodes the aggregate for classification or regression."

The model supports both Lorentz and Poincaré representations of hyperbolic space. For the Lorentz representation, the origin exponential map is given by exp L o L(v) = (c-1/2 ((v)),((v)) over(v)v), where (v) = sqrt c v. For the Poincaré representation, the exponential map is exp P o P(v) =((v)/2) over(v)v. The paper notes that the two representations are alternative coordinate choices and share the same trainable tangent centers.

HyperANFIS defines each antecedent as one geodesic region around a global rule prototype and uses dimension-normalized distance and Gaussian log-membership to compute rule activation strengths. The model also supports a generalized Bell kernel. Each rule has one learnable intrinsic scale parameterized by an unconstrained logit. The third layer normalizes the log-memberships directly to obtain firing strengths, and the paper emphasizes that this construction yields one scalar activation for each sample-rule pair. It therefore represents a global geodesic fuzzy rule rather than a product of independent coordinate-wise memberships.

For consequent aggregation, HyperANFIS expresses sample i relative to rule prototype r through the intrinsic local coordinate and constructs a first-order rule consequent using affine parameters. The fifth layer aggregates the manifold-valued consequents through their weighted Fréchet mean, computed by a fixed number of differentiable Karcher refinements. For classification, trainable class tangents gk define class prototypes and the class logits are s ik = -d M c(m M i, c M k) squared. For regression, the prediction is obtained in the origin tangent chart.

The experimental evaluation compares HyperANFIS with conventional ANFIS, which served as the primary Euclidean baseline, and four representative neuro-fuzzy models: FSRE-AdaTSK, FCM-ANFIS, IT2-ANFIS, and PSO-ANFIS. The five datasets used are Spambase, Car, Zoo, WDBC, and NSL-KDD, covering email spam detection, vehicle acceptability evaluation, animal category classification, breast cancer diagnosis, and network intrusion detection.

The results show that HyperANFIS outperforms conventional ANFIS and the representative neuro-fuzzy baselines on all five datasets. Specifically, compared with conventional ANFIS, HyperANFIS improves the average accuracy, Macro-F1 score, and Recall by 0.0473, 0.0706, and 0.0638, respectively. The improvements are particularly pronounced on Zoo and WDBC: On the Zoo dataset, HyperANFIS improves accuracy, Macro-F1, and Recall by 11.11, 13.11, and 0.1071 percentage points, respectively; on WDBC, the corresponding improvements reach 7.02, 7.76, and 0.0877 percentage points. The paper attributes these gains to the fact that Zoo exhibits a natural hierarchical taxonomic structure, whereas WDBC contains complex and nonuniform diagnostic relationships among its features, providing empirical support for the effectiveness of negative-curvature geometry in organizing nonuniform and hierarchical rule relationships.

The interpretability analysis on the WDBC dataset reveals that HyperANFIS preserves the ability of classical ANFIS to express its inference process through interpretable IF-THEN rules. The learned rules associate malignancy with increased nuclear size and contour irregularity, whereas benign patterns are generally characterized by smaller nuclear size and lower concavity, which are consistent with established findings in breast nuclear morphometry.

Comparing rule characteristics, the paper finds that classical ANFIS assigns 42.98% of the validation samples to R2 as their dominant rule, whereas the dominant coverage of the second most frequently selected rule drops to 10.53%, while HyperANFIS distributes dominant coverage across five principal rules: R4 (26.32%), R10 (21.05%), R2 (14.91%), R6 (12.28%), and R9 (12.28%). Furthermore, classical ANFIS learns 11 benign prototypes but only one malignant prototype, resulting in a substantial imbalance in class representation, whereas HyperANFIS learns seven benign and five malignant prototypes, whose distribution is more coherent. The mean off-diagonal cosine similarity between rule activation vectors increases from 0.006 to 0.432, and the mean entropy-derived effective number of active rules increases from 1.14 for classical ANFIS to 6.67 for HyperANFIS. The paper concludes that "classical ANFIS consequently exhibits an almost winner-take-all inference pattern dominated by a single rule, whereas HyperANFIS uses one or more principal rules to guide the inference, with the remaining rules providing supplementary evidence."

The paper concludes that hyperbolic geometry enables the learned IF-THEN rules to provide more complete class representations, stronger inter-rule cooperation, and a more trustworthy inference process, and that HyperANFIS achieves higher accuracy and more trustworthy predictions while producing higher-quality rules. The significance of the work extends beyond the geometric reformulation and performance improvement of classical ANFIS by providing a new geometric perspective for improving other ANFIS-family models and, more broadly, intrinsically interpretable models.

Improvements to AI systems:

1. Hierarchical data handling: Replace Euclidean-space rule antecedents with hyperbolic-space geodesic regions, enabling the system to model tree-like, hierarchical, or non-uniform data structures (e.g., taxonomies, network topologies) more efficiently with lower-dimensional embeddings.

2. Balanced rule utilization: Replace winner-take-all inference (where one rule dominates) with a distributed firing-strength mechanism that activates multiple complementary rules (effective rule count increased from 1.1 to 6.7), improving robustness and reducing over-reliance on a single prototype.

3. Class-balanced prototype learning: Learn symmetric numbers of prototypes per class (e.g., 7 benign + 5 malignant vs. 11 benign + 1 malignant in classical ANFIS), preventing under-representation of minority classes and improving recall and macro-F1 scores.

4. Geodesic distance-based membership: Use intrinsic hyperbolic distances (Lorentz or Poincaré) instead of Euclidean coordinate-wise products for rule activation, capturing non-linear relationships between features and rule centers more accurately.

5. Fréchet-mean aggregation: Aggregate rule consequents via a differentiable weighted Fréchet mean on the manifold (with Karcher refinements), rather than simple weighted sums, preserving geometric structure during inference.

6. Interpretable rule quality: Generate IF-THEN rules that are more credible and clinically consistent (e.g., matching known breast cancer morphometry), with lower inter-rule redundancy (cosine similarity 0.006 → 0.432) and higher rule diversity, improving trust in model decisions.

What the improved AI system can do:

• Achieve higher accuracy, macro-F1, and recall on datasets with hierarchical or non-uniform structures (e.g., +11.11% accuracy on Zoo, +7.02% on WDBC) compared to Euclidean ANFIS baselines.

• Provide interpretable, class-balanced rules that explain predictions in a way that aligns with domain knowledge (e.g., medical diagnostics, network intrusion detection).

• Scale to low-dimensional embeddings for high-dimensional hierarchical data, reducing memory and computation costs while maintaining performance.

• Support both classification and regression tasks with a unified hyperbolic architecture, and be adaptable to other neuro-fuzzy or intrinsically interpretable models.

The adaptive neuro-fuzzy inference system (ANFIS) is an interpretable reasoning framework capable of generating explicit IF-THEN fuzzy rules, making it suitable for tasks requiring transparent reasoning. However, existing ANFIS models generally construct rule antecedents and perform inference in Euclidean space, limiting their representational capacity and predictive performance. To address this issue, we propose Hyperbolic ANFIS (HyperANFIS), a hyperbolic extension of ANFIS. HyperANFIS preserves the fuzzy semantics and core architecture of conventional ANFIS while performing rule-prototype learning, rule activation, and consequent aggregation in hyperbolic space. It also retains the ability to generate interpretable IF-THEN rules. By exploiting the representational properties of hyperbolic geometry, HyperANFIS strengthens the fuzzy inference process, thereby improving predictive accuracy, inter-rule collaboration, and the credibility of its interpretable rules. Experimental results show that HyperANFIS consistently outperforms the standard ANFIS baseline and various ANFIS variants across all datasets, while also generating higher-quality fuzzy rules.