A typical criticism of Gaussian processes is their unfavourable scaling in both compute and memory requirements. Sparse variational Gaussian processes based on inducing variables are commonly used to scale Gaussian processes to large dataset sizes; their inherent compute and memory requirements are dominated by the number of inducing variables used. However, in practise sparse GPs are still limited by the number of datapoints and the number of inducing points one can use to perform matrix operations, making it again challenging to model large complex datasets. In this work we propose a new class of inter-domain variational GP, constructed by projecting the GP onto a set of compactly supported B-Spline basis functions. The key benefit of our approach is that the compact support of the B-Spline basis admits the use of sparse linear algebra to significantly speed up matrix operations and drastically reduce the memory footprint.

Bayesian inference in non-linear dynamical systems seeks to find good posterior approximations of a latent state given a sequence of observations. Gaussian filters and smoothers, including the (extended/unscented) Kalman filter/smoother, which are commonly used in engineering applications, yield Gaussian posteriors on the latent state. While they are computationally efficient, they are often criticised for their crude approximation of the posterior state distribution. In this paper, we address this criticism by proposing a message passing scheme for iterative state estimation in non-linear dynamical systems, which yields more informative (Gaussian) posteriors on the latent states. Our message passing scheme is based on expectation propagation (EP). We prove that classical Rauch–Tung–Striebel (RTS) smoothers, such as the extended Kalman smoother (EKS) or the unscented Kalman smoother (UKS), are special cases of our message passing scheme. Running the message passing scheme more than once can lead to significant improvements of the classical RTS smoothers, so that more informative state estimates can be obtained. We address potential convergence issues of EP by generalising our state estimation framework to damped updates and the consideration of general alpha-divergences.

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a different line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors. Starting from first principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning.

Dynamic time warping (DTW) is a useful method for aligning, comparing and combining time series, but it requires them to live in comparable spaces. In this work, we consider a setting in which time series live on different spaces without a sensible ground metric, causing DTW to become ill-defined. To alleviate this, we propose Gromov dynamic time warping (GDTW), a distance between time series on potentially incomparable spaces that avoids the comparability requirement by instead considering intra-relational geometry. We demonstrate its effectiveness at aligning, combining and comparing time series living on incomparable spaces. We further propose a smoothed version of GDTW as a differentiable loss and assess its properties in a variety of settings, including barycentric averaging, generative modeling and imitation learning.