Wil O. C. Ward

Wil O. C. Ward, Tom Ryder, Dennis Prangle et al.

Latent force models are systems whereby there is a mechanistic model describing the dynamics of the system state, with some unknown forcing term that is approximated with a Gaussian process. If such dynamics are non-linear, it can be difficult to estimate the posterior state and forcing term jointly, particularly when there are system parameters that also need estimating. This paper uses black-box variational inference to jointly estimate the posterior, designing a multivariate extension to local inverse autoregressive flows as a flexible approximater of the system. We compare estimates on systems where the posterior is known, demonstrating the effectiveness of the approximation, and apply to problems with non-linear dynamics, multi-output systems and models with non-Gaussian likelihoods.

Wil O C Ward, Mauricio A Álvarez

This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full GP regression and generated paths are demonstrated to be indistinguishable from GP samples. We show that the approach extends easily to non-linear dynamics and discuss extensions to which the approach can be easily applied.

Mauricio A. Álvarez, Wil O. C. Ward, Cristian Guarnizo

The paper introduces a non-linear version of the process convolution formalism for building covariance functions for multi-output Gaussian processes. The non-linearity is introduced via Volterra series, one series per each output. We provide closed-form expressions for the mean function and the covariance function of the approximated Gaussian process at the output of the Volterra series. The mean function and covariance function for the joint Gaussian process are derived using formulae for the product moments of Gaussian variables. We compare the performance of the non-linear model against the classical process convolution approach in one synthetic dataset and two real datasets.

Jinming Duan, Wil OC Ward, Luke Sibbett et al.

Second order total variation (SOTV) models have advantages for image reconstruction over their first order counterparts including their ability to remove the staircase artefact in the reconstructed image, but they tend to blur the reconstructed image. To overcome this drawback, we introduce a new Tensor Weighted Second Order (TWSO) model for image reconstruction. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the SOTV Hessian matrix and the anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can thus be solved efficiently. The proposed method is compared with a range of state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, Euler's elastica, etc. Numerical experimental results of the method on both synthetic and real images from the Berkeley database BSDS500 demonstrate that the proposed method eliminates both the staircase and blurring effects and outperforms the existing approaches for image inpainting and denoising applications.