Abstract
This paper is concerned with a state-space approach to deep Gaussian process (DGP) regression. We construct the DGP by hierarchically putting transformed Gaussian process (GP) priors on the length scales and magnitudes of the next level of Gaussian processes in the hierarchy. The idea of the state-space approach is to represent the DGP as a non-linear hierarchical system of linear stochastic differential equations (SDEs), where each SDE corresponds to a conditional GP. The DGP regression problem then becomes a state estimation problem, and we can estimate the state efficiently with sequential methods by using the Markov property of the state-space DGP. The computational complexity scales linearly with respect to the number of measurements. Based on this, we formulate state-space MAP as well as Bayesian filtering and smoothing solutions to the DGP regression problem. We demonstrate the performance of the proposed models and methods on synthetic non-stationary signals and apply the state-space DGP to detection of the gravitational waves from LIGO measurements.
Highlights
Gaussian processes (GP) are popular models for probabilistic non-parametric regression, especially in the machine learning field (Rasmussen and Williams 2006)
We have proposed a state-space approach to deep Gaussian process (DGP) regression
The DGP is formulated as a cascaded collection of conditional Gaussian processes (GPs)
Summary
Gaussian processes (GP) are popular models for probabilistic non-parametric regression, especially in the machine learning field (Rasmussen and Williams 2006). Damianou and Lawrence (2013) construct deep Gaussian processes (DGPs) by feeding the outputs of GPs to another layer of GPs as (transformed) inputs. The posterior inference requires complicated approximations and does not scale well with a large number of measurements (Salimbeni and Deisenroth 2017a). Another commonly used non-stationary GP construction is to have input-dependent covariance function hyperparameters, so that the resulting covariance function is non-stationary (Sampson and Guttorp 1992; Higdon et al 1999; Paciorek and Schervish 2004). One can parametrize the lengthscale as a function of time This method grants GPs the capability of changing behaviour depending on the input. One needs to be careful to ensure that the construction leads to valid
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