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AI Research
arXiv CS.AI·May 26·5 min read
LAPLEX: The FFT of Learnable Laplace Kernels
TL;DR
Deep learning often faces a trade-off between fixed geometry with exact computations and adaptive geometry with dense parameters. LAPLEX is introduced as a class of trainable Laplace-kernel operators that allows for efficient matrix operations using a small set of learnable parameters.
✦ Why It Matters
Engineers can leverage LAPLEX for efficient high-dimensional data processing in deep learning applications without incurring high storage costs.
Key Takeaways
- Utilize LAPLEX for efficient processing of large datasets.
- Implement trainable Laplace-kernel operators in neural networks.
- Explore high-dimensional covariance modeling without convolutional biases.
- Reduce storage costs while maintaining expressivity in deep learning models.
Full Summary
In deep learning, achieving fast linear algebra operations typically requires a compromise between using fixed geometry, like the Fourier transform, and adaptive geometry, which can lead to increased parameter density. LAPLEX addresses this issue by introducing a new class of exact, trainable Laplace-kernel operators that leverage Fast Fourier Transform (FFT) scaling.
Each LAPLEX layer is represented as a full-rank dense matrix defined by learnable coordinate anchors, allowing for efficient matrix-vector operations on very large dimensions, up to 10^9, on modern GPUs. This approach enables the creation of compact neural network layers that can interpret complex data structures without the need for convolutional biases.
The results show that LAPLEX can effectively model high-dimensional covariance structures in flattened images, preserving spatial relationships while minimizing storage requirements. This separation of expressivity from storage cost opens new avenues for data-adaptive global interactions in deep learning applications.