Source-linked AI summary
Wav-KAN: Wavelet Kolmogorov-Arnold Networks
Zavareh Bozorgasl, Hao Chen
TL;DR
MLPs face drawbacks including unclear decision reasoning, while Wav-KAN introduces wavelet functions within the KAN framework. The reported experiments show superior accuracy, faster training than Spl-KAN, efficient parameter usage, and improved interpretability.
Problem
MLPs have serious drawbacks, and their decisions may be made for reasons that are not clear to users.
Method
Wav-KAN integrates wavelet functions into the Kolmogorov-Arnold Networks framework.
Results
Wav-KAN achieves superior accuracy, faster training speeds than Spl-KAN, more efficient parameter usage, and improved model interpretability.
Takeaways & Limitations
Wav-KAN is presented as a powerful and versatile model for developing interpretable neural networks.
Abstract
from arXiv · showhide
In this paper, we introduce Wav-KAN, an innovative neural network architecture that leverages the Wavelet Kolmogorov-Arnold Networks (Wav-KAN) framework to enhance interpretability and performance. Traditional multilayer perceptrons (MLPs) and even recent advancements like Spl-KAN face challenges related to interpretability, training speed, robustness, computational efficiency, and performance. Wav-KAN addresses these limitations by incorporating wavelet functions into the Kolmogorov-Arnold network structure, enabling the network to capture both high-frequency and low-frequency components of the input data efficiently. Wavelet-based approximations employ orthogonal or semi-orthogonal basis and maintain a balance between accurately representing the underlying data structure and avoiding overfitting to the noise. While continuous wavelet transform (CWT) has a lot of potentials, we also employed discrete wavelet transform (DWT) for multiresolution analysis, which obviated the need for recalculation of the previous steps in finding the details. Analogous to how water conforms to the shape of its container, Wav-KAN adapts to the data structure, resulting in enhanced accuracy, faster training speeds, and increased robustness compared to Spl-KAN and MLPs. Our results highlight the potential of Wav-KAN as a powerful tool for developing interpretable and high-performance neural networks, with applications spanning various fields. This work sets the stage for further exploration and implementation of Wav-KAN in frameworks such as PyTorch and TensorFlow, aiming to make wavelets in KAN as widespread as activation functions like ReLU and sigmoid in universal approximation theory (UAT). The codes to replicate the simulations are available at https://github.com/zavareh1/Wav-KAN.
I. INTRODUCTION
The introduction motivates more interpretable neural networks and presents Wav-KAN, which integrates wavelet functions into KANs to address limitations associated with MLPs and Spl-KAN.
- MLPs are widely used but are described as having drawbacks including limited interpretability and substantial parameter consumption in transformers.
- Polysemantic neurons activate for several unrelated feature types, complicating efforts to understand individual neural-network components.
- KANs use learnable univariate activation functions on edges, with nodes summing those functions, and are associated with interpretability and accuracy advantages.
- Wav-KAN replaces network weights with wavelet functions, supports arbitrary layer counts, and sums wavelet functions at nodes.
- The authors describe Wav-KAN as the first framework using wavelets in the Kolmogorov-Arnold representation theorem for arbitrary-width and arbitrary-depth networks.
- Compared with Spl-KAN and MLPs, the proposed configuration is reported as faster, more accurate, and more robust, while supporting explainable components across diverse tasks.
II. KOLMOGOROV-ARNOLD NETWORKS
KANs are presented as a neural-network design that challenges traditional MLP concepts and is grounded in the Kolmogorov-Arnold theorem.
- KANs represent a novel neural-network design that challenges traditional concepts such as the multilayer perceptron.
- Their foundation is described as the mathematical theorem developed by Kolmogorov and Arnold.
A. The Kolmogorov-Arnold Representation Theorem
The Kolmogorov-Arnold Representation Theorem states that continuous multivariate functions can be represented using sums of univariate functions. Wav-KAN uses wavelet functions as the corresponding basis within this framework.
- The theorem states that any continuous function of n variables on [0, 1]^n can be represented using univariate functions.
- This representation decomposes a multivariate function into a sum of functions of sums.
- The inner functions are univariate, so each takes a single input.
- In Wav-KAN notation, wavelet-based functions ψ_i,j(x_j) replace ϕ_i,j(x_j), and Ψ_i replaces Φ_i.
B. From Theory to Networks
KANs replace fixed node activations and linear weights with learnable univariate functions on edges, producing outputs by summing transformed inputs across layers.
- MLP formulation: For a fully connected layer, W^(l+1,l) connects n inputs to m outputs, while x^(l), x^(l+1), and b^(l+1) denote input, output, and bias vectors.The matrix entry w_i,j is the connection from node j in layer l to node i in layer l+1.
- Network composition: The conventional network composes L affine transformations and activations, whereas KAN replaces multiplication by learnable functions with distinct parameters.The MLP composition is given explicitly in equation (8).
- KAN formulation: A KAN edge function receives one input element, transforms it, and contributes its result to an output node through summation.The operator sums matrix elements row by row to form the output vector.
- MLP formulation: The standard MLP computes each layer with a linear transformation followed by a fixed nonlinear activation, updating weights and biases by backpropagation.The MLP layer uses W^(l+1,l), biases, and fixed activation functions.
- KAN formulation: KANs place learnable univariate functions on edges instead of linear weights and use node-wise summation to transform inputs.Each function can learn both activation and input transformation.
C. Why Bother with KANs?
KANs are presented as a more flexible alternative that learns functions directly involved in data relationships, with potential gains in accuracy and interpretability.
- Motivation: Learning functions directly involved in data relationships gives KANs a more nuanced capacity to adapt to data.The paper frames this flexibility as a motivation for the architecture.
- Motivation: KANs can fit complex data patterns more precisely with potentially fewer parameters.The paper lists accuracy as a principal motivation for KANs.
- Motivation: KAN interpretability follows from assigning each learned function a specific, understandable role in the network.This makes the functions easier to inspect than undifferentiated weights.
- Motivation: The paper characterizes KANs as focusing on functions rather than weights to support a richer and more intuitive form of machine learning.This is presented as the section’s broader perspective on KANs.
III. CONTINUOUS WAVELET TRANSFORM
The Continuous Wavelet Transform analyzes signals across time, scale, and translation using scaled and shifted versions of a mother wavelet.
- CWT overview: CWT analyzes a signal’s frequency content as it varies over time by examining different scales and translations.It provides localized frequency analysis rather than a single global frequency description.
- Wavelet basis: A mother wavelet is scaled and shifted as a template whose shape determines which signal features are highlighted.The transform uses this family of scaled and translated basis functions.
- Reconstruction and conditions: A valid mother wavelet must satisfy a zero-mean condition and have finite energy.These criteria are stated as requirements for the wavelet basis.
- Wavelet coefficients: The CWT coefficient C(s, τ) measures the match between the signal g(t) and the mother wavelet ψ(t) at scale s and shift τ.The scale is positive, while the shift ranges over the real numbers.
- Reconstruction and conditions: The signal can be reconstructed from its wavelet coefficients through the inverse CWT, with Cψ determined by the chosen wavelet.The admissibility condition includes finite wavelet energy and zero mean.
IV. DISCRETE WAVELET TRANSFORM
The Discrete Wavelet Transform decomposes signals into multiresolution approximation and detail components using efficient recursive filtering and reusable coefficients.
- DWT overview: DWT decomposes a signal into frequency components using a compact, non-redundant representation based on scaled and translated wavelet functions.Its basis functions are described as orthonormal.
- Wav-KAN contribution: Wav-KAN combines local detail in dense regions with broader global information in sparse regions.This is described as useful for real-world data affected by irregular sampling and dropouts.
- Multiresolution analysis: Iterative high-pass and low-pass filtering produces approximation and detail coefficients at multiple resolution levels.Each level captures different frequency components in a hierarchical representation.
- Multiresolution analysis: Approximation coefficients a_j(k) and detail coefficients d_j(k) represent the signal at decomposition level j using scaling and wavelet functions.The discrete signal is expanded with φ_j,k(n) and ψ_j,k(n).
- Reconstruction: DWT coefficients can reconstruct the original signal through the inverse DWT, with J denoting the number of decomposition levels.The text states that reconstruction preserves information.
- Efficiency: Previously computed coefficients remain reusable when generating finer-resolution details, reducing computational overhead.This property supports efficient iterative and real-time signal processing.
- Wav-KAN contribution: Wavelet resolution adapts to sampling density: dense data receives fine-detail analysis, while sparse data emphasizes broader trends.The paper applies this principle to irregular sampling and data dropouts.
V. WAV-KAN OR SPL-KAN OR MLPS?
This section compares Wav-KAN with Spl-KAN and MLPs, emphasizing wavelets’ multiresolution and sparse representations, parameter efficiency, and training-speed advantages.
- Wavelets represent multiple detail levels and decompose data into high- and low-frequency components for feature extraction.Their sparse representations can support more efficient architectures and faster training.
- Wavelet selection and edge effects are important practical considerations because the chosen wavelet can affect performance and introduce artifacts.The section also contrasts B-splines’ smooth local control with their higher-dimensional computational burden and limited multiresolution capability.
- Wavelet approximations balance fidelity to underlying structure with resistance to noise overfitting through multiresolution decomposition.They capture local and global features while isolating significant patterns from irrelevant noise.
- Spl-KAN can capture training noise, and increasing its grid count requires computationally expensive curve fitting and may produce loss jumps.Wavelets avoid these grid-related computations, while DWT can capture additional details without recalculating previous steps.
- Wav-KAN retains Spl-KAN’s interpretability and accuracy advantages over MLPs while addressing Spl-KAN’s slow training speed.The paper also reports fewer parameters than Spl-KAN in the compared hypothetical network.
- Wav-KAN uses wavelet scaling instead of an additional activation-function term, which the paper associates with faster computation.Its learnable wavelet weight adapts the mother-wavelet shape to the approximated function.
VI. SIMULATION RESULTS
The simulations evaluate Wav-KAN with several wavelet types against Spl-KAN on MNIST using repeated trials and shared training enhancements. Results indicate that wavelet choice materially affects performance, with several wavelet configurations outperforming Spl-KAN in the reported comparisons.
- The experiments did not optimize parameters to their best possible values; they were intended to demonstrate overall Wav-KAN performance.For example, the Gaussian variance was set to 1, although grid search or learning could identify better values.
- Batch normalization improved performance and accelerated training for both Wav-KAN and Spl-KAN, so it was included in both models.
- The reported comparisons use AdamW with learning rate 0.001, weight decay 10^-4, and cross-entropy loss.
- Several wavelet configurations showed superior performance to Spl-KAN, while Shannon and Bump performed less well.The paper states that wavelet choice significantly impacts KAN-model performance and robustness to MNIST noise.
- Spl-KAN achieved better training performance in the comparison, which the paper attributes to overfitting the data.
VII. CONCLUSION
Wav-KAN integrates wavelet functions into the KAN framework to improve interpretability, performance, and efficiency. The paper reports superior accuracy, faster training than Spl-KAN, and potential for broader interpretable neural-network applications.
- VII. CONCLUSION: Wav-KAN is presented as a significant advancement for interpretable neural networks, with the ability to handle high-dimensional data and provide clear insights into model behavior.
- VII. CONCLUSION: Wav-KAN integrates wavelet functions within KAN to enhance interpretability and performance through multiresolution analysis.Wavelets help capture complex data patterns while addressing limitations of MLPs and Spl-KANs.
- VII. CONCLUSION: Wav-KAN achieves superior accuracy and faster training speeds compared to Spl-KAN.These outcomes are reported as experimental results of the proposed architecture.
- VII. CONCLUSION: Wav-KAN combines wavelet transforms with the Kolmogorov-Arnold representation theorem for more efficient parameter usage and improved model interpretability.
- VII. CONCLUSION: Future work includes optimizing Wav-KAN, testing other datasets and tasks, and implementing it in PyTorch and TensorFlow.
- VII. CONCLUSION: Wav-KAN is positioned as a powerful and versatile model that combines high performance with interpretability for more transparent and efficient neural-network architectures.