Browsing by Author "Schoeman, Johannes Cornelius"

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    Degenerate Gaussian factors for probabilistic inference
    (Stellenbosch : Stellenbosch University, 2021-12) Schoeman, Johannes Cornelius; Van Daalen, Corne E.; Du Preez, Johan; Stellenbosch University. Faculty of Engineering. Dept. of Electrical Engineering.
    ENGLISH ABSTRACT: Gaussian random variables and distributions are widely used for inference across many ap- plications, where correlations within probabilistic models can be used to calculate accurate posterior distributions over unobserved variables. In the case of perfect correlation, however, the covariance matrix of a Gaussian distribution is only positive semi -definite and therefore singular. This effectively means that linear dependencies exist among the random variables and can either be a direct artefact of the constructed model or the result of machine precision limitations during ill-conditioned numerical calculations. Consequently, traditional Gaussian parametrisations and calculations involving the inverse of the covariance matrix cannot be used in these degenerate settings. In this dissertation, we propose a parametrised factor that enables accurate and automatic inference on Gaussian networks in such degenerate settings at little additional computational cost. In contrast, a common practical solution is to employ ridge regularisation, which trades accuracy for numerical stability through approximations. Other, more principled solutions in turn do not provide all the capabilities of non-degenerate parametrisations. Our factor representation is effectively a generalisation of traditional Gaussian parametrisations where the positive-definite constraint (of the covariance matrix) has been relaxed. This is achieved by representing any possible degeneracies using Dirac delta functions. To extend the capabilities of Gaussian factors to degenerate settings, we derive various statistical operations and results (such as marginalisation, multiplication and affine transfor- mations of random variables) using our parametrised factors. The computational complexity of these operations is shown to be at most O(n3). In addition, we present means for accom- modating both linear and nonlinear models as well as for performing Bayesian model compar- ison. Finally, we apply our methodology to a representative example involving recursive state estimation of cooperative mobile robots. This illustrates the advantages of computing with explicit degenerate Gaussian factors when degeneracies arise inconsistently and unpredictably. Experimental results also reveal that using our factor definition leads to shorter computation times while requiring fewer parameters when compared to existing approaches.