Robert S. Allison

Robert Allison, Anthony Stephenson, Samuel F et al.

The accurate predictions and principled uncertainty measures provided by GP regression incur O(n^3) cost which is prohibitive for modern-day large-scale applications. This has motivated extensive work on computationally efficient approximations. We introduce a new perspective by exploring robustness properties and limiting behaviour of GP nearest-neighbour (GPnn) prediction. We demonstrate through theory and simulation that as the data-size n increases, accuracy of estimated parameters and GP model assumptions become increasingly irrelevant to GPnn predictive accuracy. Consequently, it is sufficient to spend small amounts of work on parameter estimation in order to achieve high MSE accuracy, even in the presence of gross misspecification. In contrast, as n tends to infinity, uncertainty calibration and NLL are shown to remain sensitive to just one parameter, the additive noise-variance; but we show that this source of inaccuracy can be corrected for, thereby achieving both well-calibrated uncertainty measures and accurate predictions at remarkably low computational cost. We exhibit a very simple GPnn regression algorithm with stand-out performance compared to other state-of-the-art GP approximations as measured on large UCI datasets. It operates at a small fraction of those other methods' training costs, for example on a basic laptop taking about 30 seconds to train on a dataset of size n = 1.6 x 10^6.

Robert Allison, Tomasz Maciazek, Anthony Stephenson

Gaussian process ($GP$) regression is a widely used non-parametric modeling tool, but its cubic complexity in the training size limits its use on massive data sets. A practical remedy is to predict using only the nearest neighbours of each test point, as in Nearest Neighbour Gaussian Process ($NNGP$) regression for geospatial problems and the related scalable $GPnn$ method for more general machine-learning applications. Despite their strong empirical performance, the large-$n$ theory of $NNGP/GPnn$ remains incomplete. We develop a theoretical framework for $NNGP$ and $GPnn$ regression. Under mild regularity assumptions, we derive almost sure pointwise limits for three key predictive criteria: mean squared error ($MSE$), calibration coefficient ($CAL$), and negative log-likelihood ($NLL$). We then study the $L_2$-risk, prove universal consistency, and show that the risk attains Stone's minimax rate $n^{-2α/(2p+d)}$, where $α$ and $p$ capture regularity of the regression problem. We also prove uniform convergence of $MSE$ over compact hyper-parameter sets and show that its derivatives with respect to lengthscale, kernel scale, and noise variance vanish asymptotically, with explicit rates. This explains the observed robustness of $GPnn$ to hyper-parameter tuning. These results provide a rigorous statistical foundation for $NNGP/GPnn$ as a highly scalable and principled alternative to full $GP$ models.

Anthony Stephenson, Robert Allison, Edward Pyzer-Knapp

When comparing approximate Gaussian process (GP) models, it can be helpful to be able to generate data from any GP. If we are interested in how approximate methods perform at scale, we may wish to generate very large synthetic datasets to evaluate them. Naïvely doing so would cost \(\mathcal{O}(n^3)\) flops and \(\mathcal{O}(n^2)\) memory to generate a size \(n\) sample. We demonstrate how to scale such data generation to large \(n\) whilst still providing guarantees that, with high probability, the sample is indistinguishable from a sample from the desired GP.

Dominic Broadbent, Nick Whiteley, Robert Allison et al.

Existing distribution compression methods, like Kernel Herding (KH), were originally developed for unlabelled data. However, no existing approach directly compresses the conditional distribution of labelled data. To address this gap, we first introduce the Average Maximum Conditional Mean Discrepancy (AMCMD), a natural metric for comparing conditional distributions. We then derive a consistent estimator for the AMCMD and establish its rate of convergence. Next, we make a key observation: in the context of distribution compression, the cost of constructing a compressed set targeting the AMCMD can be reduced from $\mathcal{O}(n^3)$ to $\mathcal{O}(n)$. Building on this, we extend the idea of KH to develop Average Conditional Kernel Herding (ACKH), a linear-time greedy algorithm that constructs a compressed set targeting the AMCMD. To better understand the advantages of directly compressing the conditional distribution rather than doing so via the joint distribution, we introduce Joint Kernel Herding (JKH), a straightforward adaptation of KH designed to compress the joint distribution of labelled data. While herding methods provide a simple and interpretable selection process, they rely on a greedy heuristic. To explore alternative optimisation strategies, we propose Joint Kernel Inducing Points (JKIP) and Average Conditional Kernel Inducing Points (ACKIP), which jointly optimise the compressed set while maintaining linear complexity. Experiments show that directly preserving conditional distributions with ACKIP outperforms both joint distribution compression (via JKH and JKIP) and the greedy selection used in ACKH. Moreover, we see that JKIP consistently outperforms JKH.

Dominic Broadbent, Nick Whiteley, Robert Allison et al.

Existing distribution compression methods reduce dataset size by minimising the Maximum Mean Discrepancy (MMD) between original and compressed sets, but modern datasets are often large in both sample size and dimensionality. We propose Bilateral Distribution Compression (BDC), a two-stage framework that compresses along both axes while preserving the underlying distribution, with overall linear time and memory complexity in dataset size and dimension. Central to BDC is the Decoded MMD (DMMD), which quantifies the discrepancy between the original data and a compressed set decoded from a low-dimensional latent space. BDC proceeds by (i) learning a low-dimensional projection using the Reconstruction MMD (RMMD), and (ii) optimising a latent compressed set with the Encoded MMD (EMMD). We show that this procedure minimises the DMMD, guaranteeing that the compressed set faithfully represents the original distribution. Experiments show that across a variety of scenarios BDC can achieve comparable or superior performance to ambient-space compression at substantially lower cost.

Jingbo Zhao, Robert S. Allison, Margarita Vinnikov et al.

Recent research has proposed teleoperation of robotic and aerial vehicles using head motion tracked by a head-mounted display (HMD). First-person views of the vehicles are usually captured by onboard cameras and presented to users through the display panels of HMDs. This provides users with a direct, immersive and intuitive interface for viewing and control. However, a typically overlooked factor in such designs is the latency introduced by the vehicle dynamics. As head motion is coupled with visual updates in such applications, visual and control latency always exists between the issue of control commands by head movements and the visual feedback received at the completion of the attitude adjustment. This causes a discrepancy between the intended motion, the vestibular cue and the visual cue and may potentially result in simulator sickness. No research has been conducted on how various levels of visual and control latency introduced by dynamics in robots or aerial vehicles affect users' performance and the degree of simulator sickness elicited. Thus, it is uncertain how much performance is degraded by latency and whether such designs are comfortable from the perspective of users. To address these issues, we studied a prototyped scenario of a head motion controlled quadcopter using an HMD. We present a virtual reality (VR) paradigm to systematically assess the effects of visual and control latency in simulated drone control scenarios.

Jingbo Zhao, Robert S. Allison

Head gesture is a natural means of face-to-face communication between people but the recognition of head gestures in the context of virtual reality and use of head gesture as an interface for interacting with virtual avatars and virtual environments have been rarely investigated. In the current study, we present an approach for real-time head gesture recognition on head-mounted displays using Cascaded Hidden Markov Models. We conducted two experiments to evaluate our proposed approach. In experiment 1, we trained the Cascaded Hidden Markov Models and assessed the offline classification performance using collected head motion data. In experiment 2, we characterized the real-time performance of the approach by estimating the latency to recognize a head gesture with recorded real-time classification data. Our results show that the proposed approach is effective in recognizing head gestures. The method can be integrated into a virtual reality system as a head gesture interface for interacting with virtual worlds.