Design of Positive-Definite Quaternion Kernels

This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/LSP.2015.2457294Quaternion reproducing kernel Hilbert spaces (QRKHS) have been proposed recently and provide a highdimensional feature space (alternative to the real-valued multikernel approach) for general kernel-learning applications. The current challenge within quaternion-kernel learning is the lack of general quaternion-valued kernels, which are necessary to exploit the full advantages of the QRKHS theory in real-world problems. This letter proposes a novel way to design quaternionvalued kernels, this is achieved by transforming three complex kernels into quaternion ones and then combining their real and imaginary parts. Building on this general construction, our emphasis is on a new quaternion kernel of polynomial features, which is assessed in the prediction of bodysensor networks applications.F. Tobar acknowledges financial support to EPSRC grant number EP/L000776/1

Improving Sparsity in Kernel Adaptive Filters Using a Unit-Norm Dictionary

Kernel adaptive filters, a class of adaptive nonlinear time-series models, are known by their ability to learn expressive autoregressive patterns from sequential data. However, for trivial monotonic signals, they struggle to perform accurate predictions and at the same time keep computational complexity within desired boundaries. This is because new observations are incorporated to the dictionary when they are far from what the algorithm has seen in the past. We propose a novel approach to kernel adaptive filtering that compares new observations against dictionary samples in terms of their unit-norm (normalised) versions, meaning that new observations that look like previous samples but have a different magnitude are not added to the dictionary. We achieve this by proposing the unit-norm Gaussian kernel and define a sparsification criterion for this novel kernel. This new methodology is validated on two real-world datasets against standard KAF in terms of the normalised mean square error and the dictionary size.Comment: Accepted at the IEEE Digital Signal Processing conference 201

Nonlinear Control of Unmanned Aerial Vehicles : Systems With an Attitude

This thesis deals with the general problem of controlling rigid-body systems through space, with a special focus on unmanned aerial vehicles (UAVs). Several promising UAV control algorithms have been developed over the past decades, enabling truly astounding feats of agility when combined with modern sensing technologies. However, these control algorithms typically come without global stability guarantees when implemented with estimation algorithms. Such control systems work well most of the time, but when introducing the UAVs more widely in society, it becomes paramount to prove that stability is ensured regardless of how the control system is initialized.The main motivation of the research lies in providing such (almost) global stability guarantees for an entire UAV control system. We develop algorithms that are implementable in practice and for which (almost) all initial errors result in perfect tracking of a reference trajectory. In doing so, both the tracking and the estimation errors are shown to be bounded in time along (almost) all solutions of the closed-loop system. In other words, if the initialization is sound and the initial errors are small, they will remain small and decrease in time, and even if the initial errors are large, they will not increase with time.As the field of UAV control is mature, this thesis starts by reviewing some of the most promising approaches to date in Part I. The ambition is to clarify how various controllers are related, provide intuition, and demonstrate how they work in practice. These ideas subsequently form the foundation on which a new result is derived, referred to as a nonlinear filtered output feedback. This represents a diametrically different approach to the control system synthesis. Instead of a disjoint controller/estimator design, the proposed method is comprised of two controller/estimator pairs, which when combined through a special interconnection term yields a system with favorable stability properties.While the first part of the thesis deals with theoretical controller design,Part II concerns application examples, demonstrating how the theory can solve challenging problems in modern society. In particular, we consider the problem of circumnavigation for search and rescue missions and show how UAVs can gather data from radioactive sites to estimate radiation intensity

Motion-capture-based hand gesture recognition for computing and control

This dissertation focuses on the study and development of algorithms that enable the analysis and recognition of hand gestures in a motion capture environment. Central to this work is the study of unlabeled point sets in a more abstract sense. Evaluations of proposed methods focus on examining their generalization to users not encountered during system training. In an initial exploratory study, we compare various classification algorithms based upon multiple interpretations and feature transformations of point sets, including those based upon aggregate features (e.g. mean) and a pseudo-rasterization of the capture space. We find aggregate feature classifiers to be balanced across multiple users but relatively limited in maximum achievable accuracy. Certain classifiers based upon the pseudo-rasterization performed best among tested classification algorithms. We follow this study with targeted examinations of certain subproblems. For the first subproblem, we introduce the a fortiori expectation-maximization (AFEM) algorithm for computing the parameters of a distribution from which unlabeled, correlated point sets are presumed to be generated. Each unlabeled point is assumed to correspond to a target with independent probability of appearance but correlated positions. We propose replacing the expectation phase of the algorithm with a Kalman filter modified within a Bayesian framework to account for the unknown point labels which manifest as uncertain measurement matrices. We also propose a mechanism to reorder the measurements in order to improve parameter estimates. In addition, we use a state-of-the-art Markov chain Monte Carlo sampler to efficiently sample measurement matrices. In the process, we indirectly propose a constrained k-means clustering algorithm. Simulations verify the utility of AFEM against a traditional expectation-maximization algorithm in a variety of scenarios. In the second subproblem, we consider the application of positive definite kernels and the earth mover\u27s distance (END) to our work. Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. We develop a set-theoretic interpretation of ENID and propose earth mover\u27s intersection (EMI). a positive definite analog to ENID. We offer proof of EMD\u27s negative definiteness and provide necessary and sufficient conditions for ENID to be conditionally negative definite, including approximations that guarantee negative definiteness. In particular, we show that ENID is related to various min-like kernels. We also present a positive definite preserving transformation that can be applied to any kernel and can be used to derive positive definite EMD-based kernels, and we show that the Jaccard index is simply the result of this transformation applied to set intersection. Finally, we evaluate kernels based on EMI and the proposed transformation versus ENID in various computer vision tasks and show that END is generally inferior even with indefinite kernel techniques. Finally, we apply deep learning to our problem. We propose neural network architectures for hand posture and gesture recognition from unlabeled marker sets in a coordinate system local to the hand. As a means of ensuring data integrity, we also propose an extended Kalman filter for tracking the rigid pattern of markers on which the local coordinate system is based. We consider fixed- and variable-size architectures including convolutional and recurrent neural networks that accept unlabeled marker input. We also consider a data-driven approach to labeling markers with a neural network and a collection of Kalman filters. Experimental evaluations with posture and gesture datasets show promising results for the proposed architectures with unlabeled markers, which outperform the alternative data-driven labeling method

Non-parametric regression for robot learning on manifolds

Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a quaternion, the spaces of which are non-Euclidean manifolds. In robot learning, manifold-valued data are often handled by relating the manifold to a suitable Euclidean space, either by embedding the manifold or by projecting the data onto one or several tangent spaces. These approaches can result in poor predictive accuracy, and convoluted algorithms. In this paper, we propose an "intrinsic" approach to regression that works directly within the manifold. It involves taking a suitable probability distribution on the manifold, letting its parameter be a function of a predictor variable, such as time, then estimating that function non-parametrically via a "local likelihood" method that incorporates a kernel. We name the method kernelised likelihood estimation. The approach is conceptually simple, and generally applicable to different manifolds. We implement it with three different types of manifold-valued data that commonly appear in robotics applications. The results of these experiments show better predictive accuracy than projection-based algorithms.Comment: 17 pages, 15 figure