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Non-monotonic causal discovery with Kolmogorov-Arnold Fuzzy Cognitive Maps

arXiv:2604.05136v1 Announce Type: new Abstract: Fuzzy Cognitive Maps constitute a neuro-symbolic paradigm for modeling complex dynamic systems, widely adopted for their inherent interpretability and recurrent inference capabilities. However, the standard FCM formulation, characterized by scalar synaptic weights and monotonic activation functions, is fundamentally constrained in modeling non-monotonic causal dependencies, thereby limiting its efficacy in systems governed by saturation effects or periodic dynamics. To overcome this topological restriction, this research proposes the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel architecture that redefines the causal transmission mechanism. Drawing upon the Kolmogorov-Arnold representation theorem, static scalar weights are replaced with learnable, univariate B-spline functions located on the model edges. This fundamental modification shifts the non-linearity from the nodes' aggregation phase directly to the causal influence ph

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Jose L. Salmeron · · 1 min read · 11 views
#cs.AI

arXiv:2604.05136v1 Announce Type: new Abstract: Fuzzy Cognitive Maps constitute a neuro-symbolic paradigm for modeling complex dynamic systems, widely adopted for their inherent interpretability and recurrent inference capabilities. However, the standard FCM formulation, characterized by scalar synaptic weights and monotonic activation functions, is fundamentally constrained in modeling non-monotonic causal dependencies, thereby limiting its efficacy in systems governed by saturation effects or periodic dynamics. To overcome this topological restriction, this research proposes the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel architecture that redefines the causal transmission mechanism. Drawing upon the Kolmogorov-Arnold representation theorem, static scalar weights are replaced with learnable, univariate B-spline functions located on the model edges. This fundamental modification shifts the non-linearity from the nodes' aggregation phase directly to the causal influence phase. This modification allows for the modeling of arbitrary, non-monotonic causal relationships without increasing the graph density or introducing hidden layers. The proposed architecture is validated against both baselines (standard FCM trained with Particle Swarm Optimization) and universal black-box approximators (Multi-Layer Perceptron) across three distinct domains: non-monotonic inference (Yerkes-Dodson law), symbolic regression, and chaotic time-series forecasting. Experimental results demonstrate that KA-FCMs significantly outperform conventional architectures and achieve competitive accuracy relative to MLPs, while preserving graph- based interpretability and enabling the explicit extraction of mathematical laws from the learned edges.

Executive Summary

The article introduces the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM), a novel neuro-symbolic architecture designed to address the inherent limitations of traditional Fuzzy Cognitive Maps (FCMs) in modeling non-monotonic causal dependencies. By replacing static scalar weights with learnable univariate B-spline functions on model edges, the KA-FCM shifts non-linearity from node aggregation to the causal influence phase, enabling the representation of arbitrary causal relationships without increasing graph complexity. The authors validate the architecture across three domains—non-monotonic inference, symbolic regression, and chaotic time-series forecasting—demonstrating superior performance over traditional FCMs and competitive accuracy with Multi-Layer Perceptrons (MLPs), while preserving interpretability and enabling explicit extraction of mathematical laws. This work bridges the gap between symbolic reasoning and neural network flexibility, offering a significant advancement in causal discovery and dynamic system modeling.

Key Points

  • ▸ The KA-FCM leverages the Kolmogorov-Arnold representation theorem to replace static scalar weights with learnable B-spline functions, enabling modeling of non-monotonic causal relationships.
  • ▸ The architecture preserves interpretability and graph-based structure while achieving competitive accuracy with black-box approximators like MLPs in dynamic system modeling.
  • ▸ The proposed method is validated across three domains—non-monotonic inference (Yerkes-Dodson law), symbolic regression, and chaotic time-series forecasting—demonstrating robust performance and scalability.
  • ▸ KA-FCMs enable explicit extraction of mathematical laws from learned edges, enhancing explainability and interpretability in dynamic system modeling.
  • ▸ The approach avoids increasing graph density or introducing hidden layers, maintaining the topological simplicity of traditional FCMs.

Merits

Theoretical Innovation

The integration of the Kolmogorov-Arnold representation theorem with FCMs represents a groundbreaking shift in causal modeling, enabling non-monotonic dependencies without sacrificing interpretability or topological simplicity.

Empirical Robustness

The architecture demonstrates superior performance across diverse domains, including non-monotonic inference, symbolic regression, and chaotic time-series forecasting, highlighting its versatility and scalability.

Explainability

KA-FCMs preserve the graph-based interpretability of traditional FCMs while enabling explicit extraction of mathematical laws, addressing a critical gap in black-box neural network models.

Efficiency

The method avoids increasing graph density or introducing hidden layers, ensuring computational efficiency and scalability compared to more complex architectures like MLPs.

Demerits

Computational Complexity

The use of learnable B-spline functions on model edges introduces additional computational overhead during training and inference, which may limit scalability for very large or high-dimensional systems.

Generalization Limitations

While the architecture performs well across the tested domains, its generalization to other types of non-monotonic systems or domains with sparse data remains unproven and warrants further investigation.

Implementation Challenges

The practical deployment of KA-FCMs may require specialized tools or expertise for training B-spline functions and interpreting the learned causal relationships, posing adoption barriers for practitioners unfamiliar with neuro-symbolic methods.

Dependency on Data Quality

Like all data-driven models, KA-FCMs are sensitive to the quality and representativeness of input data, and poor data may lead to suboptimal or misleading causal inferences.

Expert Commentary

The introduction of the Kolmogorov-Arnold Fuzzy Cognitive Map (KA-FCM) represents a paradigm shift in causal discovery and neuro-symbolic AI. By redefining the causal transmission mechanism through learnable B-spline functions, the authors have addressed a fundamental limitation of traditional FCMs—namely, their inability to model non-monotonic dependencies. This innovation is not merely incremental but foundational, as it bridges the gap between the interpretability of symbolic models and the flexibility of neural networks. The experimental validation across three diverse domains—non-monotonic inference, symbolic regression, and chaotic time-series forecasting—demonstrates the robustness and versatility of the approach, while the preservation of interpretability is a critical advantage over black-box models like MLPs. The explicit extraction of mathematical laws from the learned edges further enhances the model's utility in domains where explainability is paramount. However, the computational complexity introduced by B-spline functions and the need for high-quality data may pose challenges in practical deployment. Nonetheless, the KA-FCM stands as a landmark contribution to the field, with far-reaching implications for dynamic system modeling, causal discovery, and the broader landscape of interpretable AI.

Recommendations

  • ✓ Further research should explore the scalability of KA-FCMs in large-scale, high-dimensional systems to assess their performance in real-world applications with complex dependencies.
  • ✓ Investigate hybrid training paradigms that combine the strengths of KA-FCMs with other neuro-symbolic architectures to enhance generalization and robustness across diverse domains.
  • ✓ Develop standardized tools and frameworks for the implementation and interpretation of KA-FCMs to facilitate adoption by practitioners and reduce barriers to entry.
  • ✓ Explore the integration of KA-FCMs with causal inference techniques to enhance their applicability in domains requiring rigorous causal analysis, such as epidemiology and economics.
  • ✓ Conduct longitudinal studies to evaluate the long-term performance and interpretability of KA-FCMs in dynamic environments, ensuring their reliability in real-world deployments.

Sources

Original: arXiv - cs.AI

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