Time series forecasting with Hahn Kolmogorov-Arnold networks

The paper proposes HaKAN, a lightweight and interpretable time series forecasting model based on Kolmogorov-Arnold Networks with Hahn polynomial activations that effectively captures global and local temporal patterns while outperforming recent state-of-the-art methods.

The Gist

Imagine you are trying to predict the weather for the next week. You have a massive notebook filled with years of daily temperature, humidity, and wind speed records. Your goal is to look at the past and guess the future.

For a long time, computers tried to do this using two main strategies:

1. The "Super-Attentive" Approach (Transformers): These models try to look at every single data point in your history and compare it to every other point to find connections. It's like a detective reading every page of a 1,000-page book to find a clue. It's powerful, but it takes forever and gets exhausted (computationally expensive) as the book gets longer.
2. The "Simple Linear" Approach (MLPs): These models are like a student who only looks at the last few pages of the book and assumes the story will continue in a straight line. They are fast, but they often miss the twists, turns, and sudden plot twists (non-linear patterns) in the story.

Enter HaKAN: The "Smart Storyteller"

The paper introduces a new model called HaKAN (Hahn Kolmogorov-Arnold Network). Think of HaKAN as a brilliant storyteller who combines the best of both worlds but uses a special, efficient toolkit.

Here is how HaKAN works, broken down into simple concepts:

1. The "Patchwork Quilt" Strategy (Patching)

Instead of looking at the data one second at a time (which is too slow) or trying to read the whole book at once (which is too heavy), HaKAN cuts the history into patches.

• Analogy: Imagine your history is a long quilt. Instead of examining every single stitch, HaKAN cuts the quilt into small, manageable squares (patches). It studies the pattern inside each square (local details) and then looks at how the squares connect to each other (global trends).

2. The "Magic Ink" (Hahn Polynomials)

This is the secret sauce. Most models use "fixed" activation functions, which are like using a standard black ink pen. No matter what you write, the ink is the same.

• The Innovation: HaKAN uses Hahn Polynomials. Think of this as magic, shape-shifting ink.
• If the data is smooth and calm, the ink flows gently. If the data is jagged and chaotic, the ink can instantly change its shape to trace the jagged lines perfectly.
• Because this "ink" is learnable, the model doesn't just memorize the past; it learns how to draw the future patterns, whether they are smooth trends or sudden spikes. This solves the problem of "spectral bias," where other models struggle to predict sudden changes.

3. The "Two-Layer Detective" (Inter- and Intra-Patch)

HaKAN has a unique brain structure with two types of detectives working together:

• The Local Detective (Intra-Patch): This detective looks inside a single patch. "Hey, look at this 16-minute window! The temperature dropped suddenly here." It catches the fine details.
• The Global Detective (Inter-Patch): This detective looks across all the patches. "I see that every Tuesday morning, the traffic spikes. Let's connect that pattern to the whole week." It catches the big picture.
• By stacking these detectives, HaKAN understands both the tiny ripples and the massive waves in the data.

4. The "Channel Independence" Rule

In real life, different things often have their own personalities. Traffic in New York might behave differently than traffic in London, even if they are in the same dataset.

• HaKAN treats each variable (channel) separately. It doesn't mix them up. It says, "Let's listen to the temperature sensor on its own, and the humidity sensor on its own," before combining their stories at the very end. This prevents one noisy variable from confusing the whole model.

5. The "Bottleneck" Funnel

After gathering all this information, HaKAN doesn't just dump it all out. It uses a bottleneck structure.

• Analogy: Imagine squeezing a huge, fluffy cloud of data through a narrow straw before letting it out as a precise prediction. This forces the model to compress the information into its most important essence, removing the fluff and preventing it from getting confused (overfitting).

Why is this a big deal?

• Speed: It's much faster than the "Super-Attentive" models because it doesn't need to compare every single data point to every other point. It uses the "Patchwork" method to skip the heavy lifting.
• Accuracy: It's more accurate than the "Simple Linear" models because its "Magic Ink" (Hahn Polynomials) can handle complex, wiggly patterns that straight lines can't.
• Efficiency: It uses fewer computer resources (parameters) to do the job, making it lightweight and easy to run.

In Summary:
HaKAN is like a highly efficient, adaptable forecaster that cuts history into manageable chunks, uses magical, shape-shifting tools to understand both tiny details and big trends, and squeezes that knowledge into a precise prediction. It's faster than the giants and smarter than the simple models, making it a new champion for predicting the future of time series data.

Technical Summary

Here is a detailed technical summary of the paper "Time Series Forecasting with Hahn Kolmogorov-Arnold Networks" (HaKAN).

1. Problem Statement

The paper addresses the challenges of multivariate long-term time series forecasting (LTSF). Existing state-of-the-art models face specific limitations:

• Transformer-based models: While effective at capturing long-range dependencies via attention mechanisms, they suffer from quadratic computational complexity ($O(L^2)$) relative to sequence length. Furthermore, their attention mechanisms are often permutation-equivariant, which contradicts the causal nature of time series data.
• MLP-based models: These offer computational efficiency and linear complexity but exhibit spectral bias, making them poor at modeling high-frequency components and complex non-linear temporal dynamics. They often rely on linear transformations that struggle with non-linear patterns.
• Standard KANs: While Kolmogorov-Arnold Networks (KANs) offer learnable activation functions to mitigate spectral bias, standard implementations using B-splines require grid discretization, leading to high parameter counts and computational overhead dependent on grid size.

The goal is to develop a model that is computationally efficient, interpretable, capable of capturing both local and global temporal patterns, and free from the spectral bias of MLPs and the quadratic complexity of Transformers.

2. Methodology: The HaKAN Framework

The authors propose HaKAN (Hahn Kolmogorov-Arnold Network), a versatile architecture that integrates channel independence, patching, and a novel KAN block parameterized by Hahn Polynomials.

Key Architectural Components:

1. Channel Independence (CI): The model processes each variable (channel) of the multivariate time series independently to preserve unique temporal dynamics, avoiding the noise introduced by channel mixing.
2. Reversible Instance Normalization (RevIN): Used to normalize input data (handling distribution shifts) and denormalize outputs to restore the original scale.
3. Patching: The time series is divided into overlapping patches (sub-series) to capture local semantic information and improve computational efficiency.
4. Patch and Position Embeddings: Patches are projected into a latent dimension $D$ and augmented with learnable positional encodings to maintain temporal order.
5. Hahn-KAN Block (Core Innovation):
   • The backbone consists of a stack of $R$ Hahn-KAN blocks.
   • Dual-Layer Structure: Each block contains two nested KAN layers:
     • Intra-Patch KAN: Focuses on local patterns within a single patch (fine-grained dynamics).
     • Inter-Patch KAN: Focuses on global patterns across different patches (long-range dependencies).
   • Hahn Polynomial Parameterization: Unlike standard KANs that use B-splines (requiring grid discretization), HaKAN uses Hahn Polynomials as the learnable activation functions.
     • Advantage: Hahn polynomials are defined on discrete domains via recurrence relations, eliminating the need for grid size parameters ($G$). This reduces complexity from $O(d \cdot G)$ to $O(d)$ (where $d$ is polynomial degree) and significantly lowers parameter count.
   • Residual Connections: Ensures training stability by allowing the block to learn incremental updates.
6. Bottleneck Structure: The flattened features pass through a bottleneck layer (Down-projection $\to$ Up-projection) to map features to the prediction horizon $T$, reducing overfitting and computational cost.

Mathematical Formulation of Activation:

The learnable univariate function $\phi_{q,p}$ in a KAN layer is parameterized as a weighted sum of Hahn polynomials:
$$\phi_{q,p}(x_p) = \sum_{r=0}^{d} \gamma_{q,p,r} P_r(x_p)$$
Where $P_r(x)$ is the $r$-th Hahn polynomial defined by recurrence relations, and $\gamma$ are learnable coefficients.

3. Key Contributions

1. Novel Framework (HaKAN): Introduction of a multivariate LTSF framework leveraging KANs with Hahn polynomial-based activation functions, offering a lightweight and interpretable alternative to Transformers and MLPs.
2. Hahn-KAN Block Design: A hierarchical architecture integrating inter-patch (global) and intra-patch (local) KAN layers to effectively capture multi-scale temporal patterns.
3. Efficiency and Performance: Demonstrates that Hahn polynomials eliminate grid dependency, achieving linear time complexity comparable to MLPs ($O(d_{in}d_{out}d)$) while retaining the expressiveness of KANs.
4. State-of-the-Art Results: Extensive experiments show HaKAN consistently outperforms recent SOTA methods (e.g., PatchTST, iTransformer, TimeKAN, S-Mamba) across multiple benchmarks.

4. Experimental Results

The model was evaluated on 8 benchmark datasets: ETTh1, ETTh2, ETTm1, ETTm2, Weather, Traffic, Electricity, and Illness.

• Performance:
  • HaKAN achieved the best MSE in 18 out of 32 cases and best MAE in 19 out of 32 cases across varying prediction horizons ($T \in \{96, 192, 336, 720\}$).
  • Notable improvements include an 8.98% reduction in average MSE on the Illness dataset and 5.28% on ETTh2 compared to baselines.
  • It excels particularly on datasets with smooth trends (ETT series) and long-term dependencies.
• Ablation Studies:
  • Basis Functions: Hahn polynomials outperformed Lucas, Chebyshev, and B-Splines.
  • Block Count: $R=5$ blocks provided the optimal balance between performance and parameter efficiency.
  • Layer Contributions: Removing the Intra-Patch layer caused the most significant performance drop, highlighting its critical role in refining local features. The Inter-Patch layer was essential for global context.
  • Patch Length: $P=16$ was found to be the optimal patch length.
• Complexity:
  • Time Complexity: $O(M \cdot R \cdot (N^2D + ND^2))$, where $N$ is the number of patches. This is significantly more efficient than Transformers ($O(L^2)$) for long sequences because $N \ll L$.
  • Space Complexity: Significantly lower than standard KANs due to the lack of grid parameters.

5. Significance and Conclusion

The paper presents a significant advancement in time series forecasting by successfully bridging the gap between the expressiveness of KANs and the efficiency of MLPs.

• Theoretical Impact: It demonstrates that orthogonal polynomial bases (Hahn) can replace spline-based activations in KANs to achieve superior parameter efficiency without sacrificing approximation capabilities.
• Practical Impact: HaKAN offers a lightweight, interpretable, and high-performing solution for real-world applications where computational resources are limited or where understanding model behavior (via learnable activation functions) is crucial.
• Limitations: The model assumes channel independence, which may limit performance on datasets with strong inter-variable correlations (e.g., Traffic data), a common trade-off in current efficient architectures.

In summary, HaKAN establishes a new paradigm for time series forecasting by utilizing the mathematical properties of Hahn polynomials to create a model that is faster than Transformers, more expressive than MLPs, and more efficient than standard KANs.