This paper introduces a novel methodology for 3D template matching that is scalable to higher-dimensional spaces and larger kernel sizes. It uses the Hilbert Maps framework to model raw pointcloud information as a continuous occupancy function, and we derive a closed-form solution to the convolution operation that takes place directly in the Reproducing Kernel Hilbert Space defining these functions. The result is a third function modeling activation values, that can be queried at arbitrary resolutions with logarithmic complexity, and by iteratively searching for high similarity areas we can determine matching candidates. Experimental results show substantial speed gains over standard discrete convolution techniques, such as sliding window and fast Fourier transform, along with a significant decrease in memory requirements, without accuracy loss. This efficiency allows the proposed methodology to be used in areas where discrete convolution is currently infeasible. As a practical example we explore the key problem in robotics of global localization, in which a vehicle must be positioned on a map using only its current sensor information, and provide comparisons with other state-of-the-art techniques in terms of computational speed and accuracy.

Mapping the occupancy level of an environment is important for a robot to navigate in unknown and unstructured environments. To this end, continuous occupancy mapping techniques which express the probability of a location as a function are used. In this work, we provide a theoretical analysis to compare and contrast the two major branches of Bayesian continuous occupancy mapping techniques---Gaussian process occupancy maps and Bayesian Hilbert maps---considering the fact that both utilize kernel functions to operate in a rich high-dimensional implicit feature space and use variational inference to learn parameters. Then, we extend the recent Bayesian Hilbert maps framework which is so far only used for stationary robots, to map large environments with moving robots. Finally, we propose convolution of kernels as a powerful tool to improve different aspects of continuous occupancy mapping. Our claims are also experimentally validated with both simulated and real-world datasets.

Gaussian Processes (GPs) provide an extremely powerful mechanism to model a variety of problems but incur an O(N 3 ) complexity in the number of data samples. Common approximation methods rely on what are often termed inducing points but still typically incur an O(NM 2 ) complexity in the data and corresponding inducing points. Using Random Fourier Feature (RFF) maps, we overcome this by transforming the problem into a Bayesian Linear Regression formulation upon which we apply a Bayesian Variational treatment that also allows learning the corresponding kernel hyperparameters, likelihood and noise parameters. In this paper we introduce an alternative method using Fourier series to obtain spectral representations of common kernels, in particular for periodic warpings, which surprisingly have a convergent, non-random form using special functions, requiring fewer spectral features to approximate their corresponding kernel to high accuracy. Using this, we can fuse the Random Fourier Feature spectral representations of common kernels with their periodic counterparts to show how they can more effectively and expressively learn patterns in time-series for both interpolation and extrapolation. This method combines robustness, scalability and equally importantly, interpretability through a symbolic declarative grammar that is both functionally and humanly intuitive — a property that is crucial for explainable decision making. Using probabilistic programming and Variational Inference we are able to efficiently optimise over these rich functional representations. We show significantly improved Gram matrix approximation errors, and also demonstrate the method in several time-series problems comparing other commonly used approaches such as recurrent neural networks.

In budget–limited multi–armed bandit (MAB) problems, thelearner’s actions are costly and constrained by a fixed budget.Consequently, an optimal exploitation policy may not be topull the optimal arm repeatedly, as is the case in other variantsof MAB, but rather to pull the sequence of different arms thatmaximises the agent’s total reward within the budget. Thisdifference from existing MABs means that new approachesto maximising the total reward are required. Given this, wedevelop two pulling policies, namely: (i) KUBE; and (ii)fractional KUBE. Whereas the former provides better performanceup to 40% in our experimental settings, the latteris computationally less expensive. We also prove logarithmicupper bounds for the regret of both policies, and show thatthese bounds are asymptotically optimal (i.e. they only differfrom the best possible regret by a constant factor).

This paper presents multi-goal planning for an autonomous blasthole drill used in open pit mining operations. Given a blasthole pattern to be drilled and constraints on the vehicle's motion and orientation when drilling, we wish to compute the best order in which to drill the given pattern. Blasthole pattern drilling is an asymmetric Traveling Salesman Problem with precedence constraints specifying that some holes must be drilled before others. We wish to find the minimum cost tour according to criteria that minimize the distance travelled satisfying the precedence and vehicle motion constraints. We present an iterative method for solving the blasthole sequencing problem using the combination of a Genetic Algorithm and motion planning simulations that we use to determine the true cost of travel between any two holes.