LAPLEX: The FFT of Learnable Laplace Kernels
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914596449681408 |
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| author | Struski, Łukasz Blazhko, Hanna Kubaty, Piotr Tabor, Jacek |
| author_facet | Struski, Łukasz Blazhko, Hanna Kubaty, Piotr Tabor, Jacek |
| contents | Fast linear algebra in deep learning usually comes with a choice: fixed geometry and exact computation, as in the Fourier transform, or adaptive geometry paid for by dense parameters, random features, or low-rank surrogates. To move beyond this trade-off, we introduce LAPLEX, a class of exact, trainable (phased) Laplace-kernel operators.
A LAPLEX layer is a typically full-rank dense matrix, implicitly defined by learnable coordinate anchors, with FFT-like scaling. Consequently, it supports trainable matrix--vector operations at vector dimensions up to $10^9$ on modern GPUs. As a neural layer, it yields compact projections and classification heads interpretable as soft, trainable routing models. The same primitive also serves as an efficient Gram operator, enabling high-dimensional covariance models on flattened images of dimension $3 \cdot 10^6$ that preserve visible spatial structure without imposing convolutional bias. These applications reflect a single principle: dense geometry can be learned without storing a dense matrix, which enables data-adaptive global interactions in regimes where ordinary dense layers are out of reach. In this sense, LAPLEX separates expressivity from storage cost: it behaves like a dense trainable matrix, but is represented and applied through a small structured set of parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24584 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | LAPLEX: The FFT of Learnable Laplace Kernels Struski, Łukasz Blazhko, Hanna Kubaty, Piotr Tabor, Jacek Machine Learning Artificial Intelligence Fast linear algebra in deep learning usually comes with a choice: fixed geometry and exact computation, as in the Fourier transform, or adaptive geometry paid for by dense parameters, random features, or low-rank surrogates. To move beyond this trade-off, we introduce LAPLEX, a class of exact, trainable (phased) Laplace-kernel operators. A LAPLEX layer is a typically full-rank dense matrix, implicitly defined by learnable coordinate anchors, with FFT-like scaling. Consequently, it supports trainable matrix--vector operations at vector dimensions up to $10^9$ on modern GPUs. As a neural layer, it yields compact projections and classification heads interpretable as soft, trainable routing models. The same primitive also serves as an efficient Gram operator, enabling high-dimensional covariance models on flattened images of dimension $3 \cdot 10^6$ that preserve visible spatial structure without imposing convolutional bias. These applications reflect a single principle: dense geometry can be learned without storing a dense matrix, which enables data-adaptive global interactions in regimes where ordinary dense layers are out of reach. In this sense, LAPLEX separates expressivity from storage cost: it behaves like a dense trainable matrix, but is represented and applied through a small structured set of parameters. |
| title | LAPLEX: The FFT of Learnable Laplace Kernels |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.24584 |