KAN (Kolmogorov-Arnold Networks) Explained

Author(s): Jack Chih-Hsu Lin

Originally published on Towards AI.

A simple and concise summary about KANs.

Every multivariate continuous function can be represented as a superposition of the two-argument addition of continuous functions of one variable.

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f(x) is a continuous function with n-dimensional input in the interval [0,1].

f(x) can be represented as the combination of two functions, φ (phi) and Φ (uppercase phi).

Inner summation: has n univariate functions φ for each input dimension and each q.

Outer summation: has 2n+1 functions Φ for each q.

Multilayer perceptrons (MLPs)

are hard to interprethave catastrophic forgettingtraining is time-consuming since it has many weightsImage from the paper+—————————–+———————————+——————————–+| Model | MLP | KAN |+—————————–+———————————+——————————–+| Learnable Edges | Linear Weights | Activation functions || Fixed Nodes | Non-linear activation functions | Sum operations || Neural Scaling Laws | Slower | Faster || Interpretability | Lower | Higher || Has Catastrophic Forgetting | Yes due to global activations | N/A due to locality of splines |+—————————–+———————————+——————————–+

By applying Kolmogorov–Arnold Representation Theorem and learnings from MLPs, KANs were developed.

KANs have only trainable non-linear activation functions (parameterized as B-splines) whereas MLPs have trainable weights/biases and fixed activation functions. Each activation function in KAN is an univariate function. The post-activations (i.e., the outputs of activation functions) are simply added together.

Left: a two-layer KAN; Right: an activation function is parameterized as… Read the full blog for free on Medium.

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Published via Towards AI