# Feature maps for the Laplacian kernel and its generalizations

arXiv: https://arxiv.org/abs/2502.15575 · HTML: https://mlsanity.com/paper/2502.15575/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 58 references that are in the map.
- First posted on arXiv 21 Feb 2025.

## Builds on (most cited first, 30 of 30)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Orthogonal Random Features](https://mlsanity.com/paper/1610.09072/) (2016)
- [On the Error of Random Fourier Features](https://mlsanity.com/paper/1506.02785/) (2015)
- [On the Similarity between the Laplace and Neural Tangent Kernels](https://mlsanity.com/paper/2007.01580/) (2020)
- [Fastfood: Approximate Kernel Expansions in Loglinear Time](https://mlsanity.com/paper/1408.3060/) (2014)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [The Unreasonable Effectiveness of Structured Random Orthogonal Embeddings](https://mlsanity.com/paper/1703.00864/) (2017)
- [Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon](https://mlsanity.com/paper/1812.11167/) (2018)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [Quadrature-based features for kernel approximation](https://mlsanity.com/paper/1802.03832/) (2018)
- [Average gradient outer product as a mechanism for deep neural collapse](https://mlsanity.com/paper/2402.13728/) (2024)
- [Gaussian Quadrature for Kernel Features](https://mlsanity.com/paper/1709.02605/) (2017)
- [On the Inconsistency of Kernel Ridgeless Regression in Fixed Dimensions](https://mlsanity.com/paper/2205.13525/) (2022)
- [Toward Large Kernel Models](https://mlsanity.com/paper/2302.02605/) (2023)
- [Improved Random Features for Dot Product Kernels](https://mlsanity.com/paper/2201.08712/) (2022)
- [Recycling Randomness with Structure for Sublinear time Kernel Expansions](https://mlsanity.com/paper/1605.09049/) (2016)
- [A Universal Trade-off Between the Model Size, Test Loss, and Training Loss of Linear Predictors](https://mlsanity.com/paper/2207.11621/) (2022)
- [Structured adaptive and random spinners for fast machine learning computations](https://mlsanity.com/paper/1610.06209/) (2016)
- [The fast committor machine: Interpretable prediction with kernels](https://mlsanity.com/paper/2405.10410/) (2024)
- [Kernel Machines Beat Deep Neural Networks on Mask-based Single-channel Speech Enhancement](https://mlsanity.com/paper/1811.02095/) (2018)
- [Orthogonal Random Features: Explicit Forms and Sharp Inequalities](https://mlsanity.com/paper/2310.07370/) (2023)
- [Correspondence of NNGP Kernel and the Matern Kernel](https://mlsanity.com/paper/2410.08311/) (2024)

## Built on by (most cited first, 1 of 1)

- [Lipschitz bounds for integral kernels](https://mlsanity.com/paper/2604.02887/) (2026)

## Abstract (arXiv)

Recent applications of kernel methods in machine learning have seen a renewed interest in the Laplacian kernel, due to its stability to the bandwidth hyperparameter in comparison to the Gaussian kernel, as well as its expressivity being equivalent to that of the neural tangent kernel of deep fully connected networks. However, unlike the Gaussian kernel, the Laplacian kernel is not separable. This poses challenges for techniques to approximate it, especially via the random Fourier features (RFF) methodology and its variants. In this work, we provide random features for the Laplacian kernel and its two generalizations: Mat'ern kernel and the Exponential power kernel. We provide efficiently implementable schemes to sample weight matrices so that random features approximate these kernels. These weight matrices have a weakly coupled heavy-tailed randomness. Via numerical experiments on real datasets we demonstrate the efficacy of these random feature maps.