# Modelling Non-Smooth Signals with Complex Spectral Structure

arXiv: https://arxiv.org/abs/2203.06997 · HTML: https://mlsanity.com/paper/2203.06997/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 14 Mar 2022.

## Builds on (most cited first, 11 of 11)

- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Manifold Gaussian Processes for Regression](https://mlsanity.com/paper/1402.5876/) (2014)
- [On Sparse variational methods and the Kullback-Leibler divergence between stochastic processes](https://mlsanity.com/paper/1504.07027/) (2015)
- [Deep Gaussian Processes for Regression using Approximate Expectation Propagation](https://mlsanity.com/paper/1602.04133/) (2016)
- [Differentiable Compositional Kernel Learning for Gaussian Processes](https://mlsanity.com/paper/1806.04326/) (2018)
- [Function-Space Distributions over Kernels](https://mlsanity.com/paper/1910.13565/) (2019)
- [Bayesian Nonparametric Kernel-Learning](https://mlsanity.com/paper/1506.08776/) (2015)
- [On kernel design for regularized LTI system identification](https://mlsanity.com/paper/1612.03542/) (2016)
- [Scalable L'evy Process Priors for Spectral Kernel Learning](https://mlsanity.com/paper/1802.00530/) (2018)
- [Regularized parametric system identification: a decision-theoretic formulation](https://mlsanity.com/paper/1710.04009/) (2017)

## Built on by (most cited first, 2 of 2)

- [Non-Gaussian Process Regression](https://mlsanity.com/paper/2209.03117/) (2022)
- [Shallow and Deep Nonparametric Convolutions for Gaussian Processes](https://mlsanity.com/paper/2206.08972/) (2022)

## Abstract (arXiv)

The Gaussian Process Convolution Model (GPCM; Tobar et al., 2015a) is a model for signals with complex spectral structure. A significant limitation of the GPCM is that it assumes a rapidly decaying spectrum: it can only model smooth signals. Moreover, inference in the GPCM currently requires (1) a mean-field assumption, resulting in poorly calibrated uncertainties, and (2) a tedious variational optimisation of large covariance matrices. We redesign the GPCM model to induce a richer distribution over the spectrum with relaxed assumptions about smoothness: the Causal Gaussian Process Convolution Model (CGPCM) introduces a causality assumption into the GPCM, and the Rough Gaussian Process Convolution Model (RGPCM) can be interpreted as a Bayesian nonparametric generalisation of the fractional Ornstein-Uhlenbeck process. We also propose a more effective variational inference scheme, going beyond the mean-field assumption: we design a Gibbs sampler which directly samples from the optimal variational solution, circumventing any variational optimisation entirely. The proposed variations of the GPCM are validated in experiments on synthetic and real-world data, showing promising results.