LAPLEX：可学习 Laplace 核的 FFT 类快速线性算子
LAPLEX: The FFT of Learnable Laplace Kernels
AI 导读
arXiv 论文提出 LAPLEX，一类精确、可训练的（相位化）Laplace 核算子，兼具 FFT 式扩展性与可学习几何结构。其层由可学习坐标锚点隐式定义的近似满秩稠密矩阵，可在现代 GPU 上支持向量维度最高 10^9 的可训练矩阵-向量运算，并可作为 Gram 算子处理维度 3·10^6 的展平图像协方差模型。该原语无需存储稠密矩阵即可学习稠密几何，将表达能力与存储成本解耦。
正文 · 原文
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.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2605.24584 [cs.LG] |
| (or arXiv:2605.24584v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24584 arXiv-issued DOI via DataCite |
Submission history
From: Piotr Kubaty [view email]
[v1]
Sat, 23 May 2026 13:48:33 UTC (17,393 KB)
[v2]
Wed, 4 Nov 2026 22:04:34 UTC (17,393 KB)
来源：arXiv cs.AI · arxiv.org