Scaling Continuous Kernels with Sparse Fourier Domain Learning (2409.09875v1)

Abstract: We address three key challenges in learning continuous kernel representations: computational efficiency, parameter efficiency, and spectral bias. Continuous kernels have shown significant potential, but their practical adoption is often limited by high computational and memory demands. Additionally, these methods are prone to spectral bias, which impedes their ability to capture high-frequency details. To overcome these limitations, we propose a novel approach that leverages sparse learning in the Fourier domain. Our method enables the efficient scaling of continuous kernels, drastically reduces computational and memory requirements, and mitigates spectral bias by exploiting the Gibbs phenomenon.

• The paper introduces CF-Convs to mitigate spectral bias and accurately capture high-frequency details by learning directly in the Fourier domain.
• It employs continuous parameterization and a novel sparse update mechanism to reduce parameter counts and accelerate training.
• Experimental results show competitive accuracy and efficiency, paving the way for scalable CNN architectures in memory-constrained applications.

Scaling Continuous Kernels with Sparse Fourier Domain Learning

The paper "Scaling Continuous Kernels with Sparse Fourier Domain Learning" addresses critical challenges in the deployment of continuous kernel representations for convolutional neural networks (CNNs). The proposed approach, Continuous Fourier Convolutions (CF-Convs), aims to reduce the computational and memory costs associated with these kernels while addressing spectral bias limitations.

Key Contributions and Insights

The authors focus on overcoming three fundamental challenges: high parameter counts, computational and memory demands, and spectral bias. The contributions of the paper are highlighted as follows:

1. Fourier Domain Learning: CF-Convs learn directly in the Fourier domain, mitigating spectral bias through leveraging the Gibbs phenomenon. This allows the model to better capture high-frequency details essential for certain tasks.
2. Parameter Efficiency: By using a continuous representation of kernels, CF-Convs avoid parameter explosion common in Fourier domain learning. This is achieved by intelligently parameterizing the kernels across different axes.
3. Sparse Updates: A novel sparse update mechanism accelerates training and reduces memory consumption, making the approach viable for large-scale applications.

Technical Approach

The authors present a detailed methodology for addressing each challenge:

• Fourier Domain Motivation: Learning in the Fourier domain can alleviate spectral bias, a limitation where neural networks favor low-frequency components, hindering the learning of high-frequency details. Fourier domain learning allows the model to exploit the Gabor limit, effectively enabling low-frequency functions in the Fourier domain to correspond to high-pass filters in the spatial domain.
• Parameterization Strategies: Several parameterization methods are explored, conditioned on different axes (e.g., $\Phi_\Theta(H, W)$, $\Phi_\Theta(H, W, C_{\text{in}, C_{\text{out})$). The choice of parameterization significantly impacts the balance between parameter count and memory usage. The $\Phi_\Theta(H, W, C_{\text{in}, C_{\text{out})$ configuration is recommended for its expressiveness and efficiency.
• Memory and Computation Efficient Techniques: The authors propose gradient checkpointing and scan operations to manage memory usage. Despite their utility, these techniques lead to impractically long training times. Therefore, sparse kernel updates are introduced, where only a subset of kernel positions is updated at each step, reducing the computational load.

Experimental Evaluation

The paper provides comprehensive experimental results using a 6-layer CNN on the Cats vs. Dogs dataset. Key findings include:

• Efficiency: The sparse update mechanism dramatically reduces training time compared to naive methods, achieving a practical balance between memory usage and computational efficiency.
• Performance: While CF-Convs haven't yet surpassed traditional $3 \times 3$ CNNs in performance, they show competitive accuracy, especially with an increased number of selected positions for sparse updates. Specifically, the $\Phi_\Theta(H, W, C_{\text{in}, C_{\text{out})$ configuration with sparse updates achieves comparable performance to smaller spatial kernels, highlighting its potential for further optimization.

Implications and Future Directions

Practically, the findings suggest that CF-Convs can be further optimized to scale to larger and more complex CNN architectures, making them suitable for tasks requiring the capture of fine-grained details. Theoretically, this work advances the understanding of continuous kernel representations and their implementation in the Fourier domain. Future avenues could explore more efficient parameterizations and fine-tuning activation functions within the Fourier domain to enhance model performance further.

Conclusion

This paper makes a significant contribution to the field by proposing a method that scales continuous kernel representations efficiently in the Fourier domain. The innovative sparse updates mechanism and Fourier-based learning mitigate memory and computational constraints and spectral bias, offering a promising new direction in neural network research. While further optimization is required, CF-Convs pave the way for more flexible and scalable models, particularly relevant for applications demanding high-frequency detail capture.