Modelling Non-Smooth Signals with Complex Spectral Structure
This work addresses a specific problem in signal processing for researchers and practitioners dealing with non-smooth signals, representing an incremental improvement over the existing GPCM.
The authors tackled the limitation of the Gaussian Process Convolution Model (GPCM) in only modeling smooth signals by redesigning it to handle non-smooth signals with complex spectral structure, introducing variants like CGPCM and RGPCM, and proposing a more effective variational inference scheme, with experiments on synthetic and real-world data showing promising results.
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.