Coffee Time Papers: Kolmogorov–Arnold Networks

This blog post is part of the series Coffee Time Papers.

Paper

https://arxiv.org/abs/2404.19756

What is the Kolmogorov-Arnold Representation Theorem?

The Kolmogorov-Arnold representation theorem is a mathematical theorem that states that any multivariate continuous function on a bounded domain can be represented as a finite composition of continuous functions of a single variable and the binary operation of addition. In simpler terms, it means that any complex function with multiple inputs can be broken down into a combination of simpler one-dimensional functions and addition.

The theorem can be illustrated with the following example:

f(x, y) = exp(sin(πx) + y^2)

This function can be represented as a composition of univariate functions and addition as follows:

f(x, y) = Φ(φ1(x) + φ2(y))

where:

Φ(z) = exp(z)
φ1(x) = sin(πx)
φ2(y) = y^2

In this representation, Φ, φ1, and φ2 are all univariate functions, and the only multivariate operation is addition. This demonstrates how the Kolmogorov-Arnold representation theorem breaks down a complex function into simpler components.

What are Kolmogorov-Arnold Networks (KANs)?

KANs are a new type of neural network inspired by the Kolmogorov-Arnold representation theorem. Unlike traditional Multi-Layer Perceptrons (MLPs) that have fixed activation functions on nodes, KANs have learnable activation functions on edges. This means that instead of having fixed weights, each weight in a KAN is replaced by a learnable function, typically parameterized as a spline.

What are the advantages of KANs over MLPs?

KANs offer several advantages over MLPs:

• Accuracy: KANs can achieve comparable or better accuracy than MLPs with fewer parameters, especially in high-dimensional function fitting tasks. This is because they can efficiently learn both the compositional structure of a function and the individual univariate functions that make it up.
• Interpretability: KANs are more transparent and easier to understand than MLPs. The learnable activation functions on edges can be visualized, allowing for a clearer understanding of how the network is making predictions.
• Scientific Discovery: KANs can be used as tools for scientific discovery. They have been used to rediscover relationships in knot theory and identify phase transitions in condensed matter physics.

How do KANs work?

KANs are based on the Kolmogorov-Arnold representation theorem, which states that any multivariate continuous function can be represented as a finite composition of continuous functions of a single variable and the addition operation. KANs parameterize these univariate functions using B-splines, which are piecewise polynomial curves that can be easily learned.

In a KAN, the nodes simply sum the incoming signals without applying any nonlinearities. The nonlinearity is introduced by the learnable activation functions on the edges. This structure allows KANs to learn complex functions while remaining interpretable.

How do KANs parameterize univariate functions using B-splines?

Parameterizing univariate functions using B-splines means representing these one-dimensional functions as a combination of B-spline basis functions. B-splines are piecewise polynomial curves defined over a set of intervals, called a knot vector. Each B-spline basis function is non-zero only within a specific range of the knot vector, and they are designed to have desirable properties like smoothness and local support.

To parameterize a univariate function using B-splines, we express it as a linear combination of B-spline basis functions:

function(x) = c1 * B1(x) + c2 * B2(x) + ... + cn * Bn(x)

where:

• c1, c2, ..., cn are the coefficients that control the shape of the function. These are the learnable parameters in a KAN.
• B1(x), B2(x), ..., Bn(x) are the B-spline basis functions.

For example, consider a simple univariate function:

f(x) = x^2

We can approximate this function using B-splines. Let’s say we choose to use cubic B-splines (degree 3) and a knot vector of [0, 1, 2, 3]. This would give us four B-spline basis functions. We can then find the coefficients c1, c2, c3, and c4 that best approximate the function f(x) = x^2 by minimizing the error between the B-spline representation and the true function.

In the context of KANs, each of the univariate functions in the Kolmogorov-Arnold representation is parameterized in this way. The network learns the optimal coefficients for each B-spline representation during training, allowing it to accurately approximate complex functions.

How do KANs learn the B-spline coefficients during training?

KANs are trained using backpropagation, a common algorithm in deep learning. Backpropagation calculates the gradient of the loss function with respect to the network’s parameters, which are then updated to minimize the loss. In the case of KANs, the parameters are the coefficients of the B-spline basis functions that define the learnable activation functions on the edges.

During training, the input data is fed forward through the network, and the output is compared to the ground truth. The difference between the predicted output and the ground truth is the loss. The backpropagation algorithm then calculates how much each parameter contributed to the loss and updates the parameters accordingly. This process is repeated iteratively until the loss converges to a minimum.

The key difference between KANs and MLPs lies in the nature of the parameters being updated. In MLPs, the parameters are linear weights, while in KANs, the parameters are the coefficients of the B-spline basis functions. This means that KANs are not just learning linear transformations, but also the shape of the nonlinear activation functions themselves.

The authors also introduce a technique called “grid extension” to improve the accuracy of KANs. This involves increasing the number of grid points used to define the B-spline basis functions, effectively making the activation functions more fine-grained. This allows KANs to better approximate complex functions.

What are the challenges of using KANs?

One of the main challenges of using KANs is that they can be slower to train than MLPs due to the increased complexity of the learnable activation functions. However, the authors suggest several ways to improve the efficiency of KANs, such as using multi-head attention and adaptive grid strategies.

What are the potential applications of KANs?

KANs have a wide range of potential applications, including:

• Scientific Discovery: Helping scientists discover new mathematical and physical laws.
• Machine Learning: Replacing MLPs in deep learning models like transformers.
• Continual Learning: Training models that can learn new tasks without forgetting old ones.

How do KANs address the curse of dimensionality?

A: The curse of dimensionality refers to the exponential increase in the number of samples needed to learn a function as the number of input dimensions increases. KANs address this issue by leveraging the compositional structure of functions. By decomposing a high-dimensional function into a composition of lower-dimensional functions, KANs can learn the function with fewer parameters and samples.

What is the difference between external and internal degrees of freedom in KANs?

In KANs, external degrees of freedom refer to the structure of the computational graph, i.e., how the nodes are connected. Internal degrees of freedom refer to the parameters of the learnable activation functions on the edges. Both types of degrees of freedom are important for the performance of KANs. External degrees of freedom allow KANs to learn the compositional structure of functions, while internal degrees of freedom allow them to learn the individual univariate functions.

How can KANs be made more interpretable?

The authors propose several techniques to make KANs more interpretable:

• Sparsification: Using L1 and entropy regularization to encourage the network to use fewer active activation functions.
• Pruning: Removing unimportant neurons from the network.
• Symbolization: Replacing learned activation functions with known symbolic functions (e.g., sin, cos, exp).
• Visualization: Using transparency to visualize the magnitude of the activation functions, making it easier to identify important features.

How can KANs be used for scientific discovery?

A: KANs can be used for scientific discovery in both supervised and unsupervised settings. In the supervised setting, KANs can be trained to predict a target variable from a set of input variables. The learned activation functions can then be analyzed to gain insights into the relationship between the variables. In the unsupervised setting, KANs can be used to discover relationships between variables without the need for a target variable. This is done by training the KAN to distinguish between real and fake data samples.

What are the limitations of KANs?

One of the main limitations of KANs is that they can be slow to train compared to MLPs. However, the authors suggest several ways to improve the efficiency of KANs. Another limitation is that the interpretability of KANs can be challenging when the network is large or complex. However, the authors propose several techniques to make KANs more interpretable.