LAPLEX: The FFT of Learnable Laplace Kernels

• URL: http://arxiv.org/abs/2605.24584v1
• Date: Sat, 23 May 2026 13:48:33 GMT
• Title: LAPLEX: The FFT of Learnable Laplace Kernels
• Abstract summary: We introduce LAPLEX, a class of exact, trainable (phased) Laplace-kernel operators.<n>A LAPLEX layer is a typically full-rank dense matrix, implicitly defined by learnable coordinate anchors, with FFT-like scaling.<n>As a neural layer, it yields compact projections and classification heads interpretable as soft, trainable routing models.<n>These applications reflect a single principle: dense geometry can be learned without storing a dense matrix.
• Score: 9.114051379014429
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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