Kolmogorov-Arnold Networks (KAN)

KAN: Kolmogorov-Arnold Networks paperA promising alternative to Multilayer Perceptron (MLP) called Kolmogorov-Arnold Networks (KAN) is proposed. The name KAN comes from the memory of two great late mathematicians, Andrey Kolmogorov and Vladimir Arnold. The design of MLP is inspired by the universal approximation theorem, while the design of KAN is inspired by the Kolmogorov-Arnold representation theorem.

Kolmogorov-Arnold networks are a new type of neural network that takes a fundamentally different approach to learning than MLPs. While MLPs have fixed activation functions on the nodes (or "neurons"), KANs have learnable activation functions on the edges (or "weights"). This seemingly simple change has profound effects on the performance and interpretability of the network.

In KAN, each weight parameter is replaced by a univariate function, usually parameterized as a spline function. Therefore, KAN has no linear weights at all. The nodes in KAN simply sum the input signals without applying any nonlinearity.

How KAN works

The core of KAN is to learn the combinatorial structure (external degrees of freedom) and univariate functions (internal degrees of freedom) of a given problem. This allows KAN to not only learn features like MLP, but also optimize these learned features very accurately.

KANs take advantage of the strengths of both splines and MLPs while avoiding their weaknesses. Splines are accurate for low-dimensional functions and can be easily adjusted locally, but suffer from the curse of dimensionality. MLPs, on the other hand, are better at exploiting combinatorial structure but have difficulty optimizing univariate functions. By combining these two approaches, KANs can learn and accurately represent complex functions more efficiently than either splines or MLPs alone.

The Impact of KAN

The introduction of the Kolmogorov-Arnold network has two implications:

1. Improve accuracy: In tasks such as data fitting and solving partial differential equations (PDEs), KANs show comparable or better accuracy than larger MLPs. This suggests that KANs can produce more efficient and accurate models in various fields.
2. Enhanced explainability: KANs are designed to be more interpretable than MLPs. Learnable activation functions can be visualized and interacted with, giving users insight into the inner workings of the model. This interpretability is particularly valuable in fields like healthcare, where understanding the model’s decision-making process is critical.

Integrating KAN into large language models could lead to significant advances in generative AI, potentially surpassing existing neural network architectures in terms of efficiency, interpretability, few-shot learning, knowledge representation, and multimodal learning.

References

【1】KAN: Kolmogorov–Arnold Networks

【2】What is the new Neural Network Architecture?(KAN) Kolmogorov-Arnold Networks Explained