Gaussian Process Kolmogorov-Arnold Networks (2407.18397v2)

Abstract: In this paper, we introduce a probabilistic extension to Kolmogorov Arnold Networks (KANs) by incorporating Gaussian Process (GP) as non-linear neurons, which we refer to as GP-KAN. A fully analytical approach to handling the output distribution of one GP as an input to another GP is achieved by considering the function inner product of a GP function sample with the input distribution. These GP neurons exhibit robust non-linear modelling capabilities while using few parameters and can be easily and fully integrated in a feed-forward network structure. They provide inherent uncertainty estimates to the model prediction and can be trained directly on the log-likelihood objective function, without needing variational lower bounds or approximations. In the context of MNIST classification, a model based on GP-KAN of 80 thousand parameters achieved 98.5% prediction accuracy, compared to current state-of-the-art models with 1.5 million parameters.

• The paper introduces GP-KAN, integrating Gaussian Process neurons into Kolmogorov-Arnold Networks to enhance non-linear modelling and uncertainty quantification.
• The paper demonstrates analytical uncertainty propagation with univariate GPs, reducing computational complexity without resorting to variational approximations.
• The paper achieves 98.5% accuracy on MNIST with only 80,000 parameters, showcasing significant efficiency and scalability compared to larger models.

GP-KAN: Gaussian Process Kolmogorov-Arnold Networks

The paper "GP-KAN: Gaussian Process Kolmogorov-Arnold Networks" by Andrew Siyuan Chen introduces an innovative probabilistic extension to the Kolmogorov-Arnold Networks (KANs) by integrating Gaussian Processes (GPs) as non-linear neurons. This framework is referred to as GP-KAN and represents a significant enhancement in the capability of neural networks to model complex non-linear relationships while maintaining analytical tractability of uncertainty propagation.

Core Contributions

1. Introduction of GP Neurons:
   • The authors replace the traditional B-Spline non-linear neurons of KANs with Gaussian Processes, forming a new type of deep GP model. This integration allows each neuron within the network to model complex non-linear functions probabilistically.
2. Analytical Uncertainty Propagation:
   • A novel approach is introduced to handle the output distribution of a GP treating the input of another GP by considering the function inner product of a GP sample with the input distribution. This maintains Gaussian nature on input and output, facilitating exact analytical uncertainty propagation.
3. Efficiency and Scalability:
   • GP-KAN employs only univariate GPs, simplifying the analysis. The propagation of elements (Gaussian-distributed and independent) from layer to layer avoids the need for variational lower bounds or other approximations, resulting in reduced computational complexity $\mathcal{O}(n)$ for $n$ intermediate elements.
4. Performance on MNIST Dataset:
   • A model based on GP-KAN was able to achieve a remarkable test accuracy of 98.5% on the MNIST dataset with only 80,000 parameters. This performance is notable compared to state-of-the-art models with 1.5 million parameters.

Key Technical Details

• Univariate GPs as Basis Functions:
  • Univariate GPs serve as the non-linear neurons, avoiding the curse of dimensionality inherent to GPs. The output of a GP neuron is Gaussian-distributed, and analytical expressions for mean and variance can be derived (Equation $$\mathbb{E}\left[\tilde{y}\right], \text{Var}\left[\tilde{y}\right]$$).
• Fully-Connected GP Layers:
  • Extends the idea of feed-forward layers to include GP neurons with analytical propagation of Gaussian-distributed inputs and outputs. This maintains computational feasibilities similar to traditional neural networks while providing inherent uncertainty estimates.
• Convolutional GP Layers (ConvGP):
  • Adaptation of convolutional layers to GP-KAN using the im2col technique. This structured transformation ensures GP neurons capture translation-invariant features effectively, maintaining the Gaussian properties.
• Normalization:
  • A novel bijective mapping ensures the mean and variance of Gaussian random variables remain bounded, stabilizing the training process (Equation $\gamma: \mathcal{G} \rightarrow \mathcal{G}'$).

Practical and Theoretical Implications

The integration of GP into KANs, resulting in GP-KAN, provides a dual advantage of robust non-linear modelling and a full probabilistic framework. This enables deep networks to model data more efficiently, with fewer parameters yet retaining or enhancing accuracy. The analytical tractability of uncertainty propagation allows GP-KAN to provide reliable predictions with associated confidence intervals, which is paramount in critical applications where uncertainty quantification is essential. Additionally, the model's inherent ability to forego explicit activation functions suggests adaptability across diverse datasets with varying output nature.

Future Developments

Potential areas for extending this work include:

• Scalability to Larger Datasets:
  • Further performance evaluations on large-scale datasets and exploring methods to scale GP-KAN while managing computational resources effectively.
• Advanced Layer Designs:
  • Developing more complex layers and integration schemes that leverage the probabilistic interpretation of GPs to capture deeper hierarchical relationships in data.
• Real-world Applicability:
  • Applying GP-KAN to various real-world problems, particularly those requiring high reliability and certainty in predictions, such as medical diagnosis, financial forecasting, and autonomous systems.

In conclusion, the GP-KAN model proposed in this paper represents a substantial step forward in the development of probabilistic neural networks. By effectively combining the strength of Gaussian Processes with the structure of Kolmogorov-Arnold Networks, the model achieves impressive results on benchmark tasks while maintaining a low parameter count and providing inherent uncertainty estimates. This approach holds significant promise for future advancements in neural network architectures and their applications.