# Hida-Mat'ern Kernel

arXiv: https://arxiv.org/abs/2107.07098 · HTML: https://mlsanity.com/paper/2107.07098/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 32 references that are in the map.
- First posted on arXiv 15 Jul 2021.

## Builds on (most cited first, 10 of 10)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Streaming Sparse Gaussian Process Approximations](https://mlsanity.com/paper/1705.07131/) (2017)
- [Bayesian Semi-supervised Learning with Graph Gaussian Processes](https://mlsanity.com/paper/1809.04379/) (2018)
- [Generalized Spectral Kernels](https://mlsanity.com/paper/1506.02236/) (2015)
- [Hierarchical Gaussian Process Priors for Bayesian Neural Network Weights](https://mlsanity.com/paper/2002.04033/) (2020)
- [Linear-time inference for Gaussian Processes on one dimension](https://mlsanity.com/paper/2003.05554/) (2020)
- [Band-Limited Gaussian Processes: The Sinc Kernel](https://mlsanity.com/paper/1909.07279/) (2019)
- [Infinite-Horizon Gaussian Processes](https://mlsanity.com/paper/1811.06588/) (2018)

## Built on by (most cited first, 5 of 5)

- [Learning Time-Varying Multi-Region Brain Communications via Scalable Markovian Gaussian Processes](https://mlsanity.com/paper/2407.00397/) (2024)
- [A Gaussian Process-based Streaming Algorithm for Prediction of Time Series With Regimes and Outliers](https://mlsanity.com/paper/2406.00570/) (2024)
- [Modeling Neural Activity with Conditionally Linear Dynamical Systems](https://mlsanity.com/paper/2502.18347/) (2025)
- [Optimizing Orbital Parameters of Satellites for a Global Quantum Network](https://mlsanity.com/paper/2603.02480/) (2026)
- [Sequential Inference for Gaussian Processes: A Signal Processing Perspective](https://mlsanity.com/paper/2604.28163/) (2026)

## Abstract (arXiv)

We present the class of Hida-Mat'ern kernels, which is the canonical family of covariance functions over the entire space of stationary Gauss-Markov Processes. It extends upon Mat'ern kernels, by allowing for flexible construction of priors over processes with oscillatory components. Any stationary kernel, including the widely used squared-exponential and spectral mixture kernels, are either directly within this class or are appropriate asymptotic limits, demonstrating the generality of this class. Taking advantage of its Markovian nature we show how to represent such processes as state space models using only the kernel and its derivatives. In turn this allows us to perform Gaussian Process inference more efficiently and side step the usual computational burdens. We also show how exploiting special properties of the state space representation enables improved numerical stability in addition to further reductions of computational complexity.