Non-monotonic causal discovery with Kolmogorov-Arnold Fuzzy Cognitive Maps
Summary: arXiv:2604.05136v1 Announce Type: new
Abstract: Fuzzy Cognitive Maps (FCMs) 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.
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.
The proposed architecture allows for the modeling of arbitrary, non-monotonic causal relationships without increasing the graph density or introducing hidden layers. This innovation is particularly significant for applications that involve complex interactions and dependencies, where traditional methods may fall short.
Key Features of the KA-FCM
- Learnable Functions: The introduction of univariate B-spline functions enhances the flexibility of the model, enabling it to capture complex non-linear relationships.
- Non-monotonic Relationships: KA-FCM can effectively model causal dependencies that are not strictly increasing or decreasing, which is crucial for accurately representing real-world systems.
- Graph-based Interpretability: Despite the added complexity, the architecture maintains a degree of interpretability, allowing users to extract meaningful insights from the learned causal relationships.
Experimental Validation
The proposed architecture is validated against both baselines, specifically the standard FCM trained with Particle Swarm Optimization, and universal black-box approximators such as Multi-Layer Perceptron (MLP) across three distinct domains:
- Non-monotonic Inference: The Yerkes-Dodson law is utilized to demonstrate the model’s capability in handling non-linear relationships.
- Symbolic Regression: KA-FCM excels in deriving mathematical representations from data, showcasing its effectiveness in regression tasks.
- Chaotic Time-Series Forecasting: The architecture proves its reliability in predicting chaotic systems, which are inherently complex and challenging to model.
Experimental results indicate that KA-FCMs significantly outperform conventional architectures, providing enhanced accuracy while preserving interpretability. In many cases, they achieve competitive performance relative to MLPs, making them a viable alternative in the field of causal discovery.
Conclusion
The introduction of the Kolmogorov-Arnold Fuzzy Cognitive Map marks a significant advancement in the modeling of complex dynamic systems. By enabling the representation of non-monotonic causal relationships through innovative adjustments to the FCM framework, this research opens new avenues for exploration in causal inference and dynamic systems modeling.