Variational methods for geometric statistical inference
Estimating multiple geometric shapes such as tracks or surfaces creates significant mathematical challenges particularly in the presence of unknown data association. In particular, problems of this type have two major challenges. The first is typically the object of interest is infinite dimensional whilst data is finite dimensional. As a result the inverse problem is ill-posed without regularization. The second is the data association makes the likelihood function highly oscillatory.\ud \ud The focus of this thesis is on techniques to validate approaches to estimating problems in geometric statistical inference. We use convergence of the large data limit as an indicator of robustness of the methodology. One particular advantage of our approach is that we can prove convergence under modest conditions on the data generating process. This allows one to apply the theory where very little is known about the data. This indicates a robustness in applications to real world problems.\ud \ud The results of this thesis therefore concern the asymptotics for a selection of statistical inference problems. We construct our estimates as the minimizer of an appropriate functional and look at what happens in the large data limit. In each case we will show our estimates converge to a minimizer of a limiting functional. In certain cases we also add rates of convergence.\ud \ud The emphasis is on problems which contain a data association or classification component. More precisely we study a generalized version of the k-means method which is suitable for estimating multiple trajectories from unlabeled data which combines data association with spline smoothing. Another problem considered is a graphical approach to estimating the labeling of data points. Our approach uses minimizers of the Ginzburg-Landau functional on a suitably defined graph.\ud \ud In order to study these problems we use variational techniques and in particular I-convergence. This is the natural framework to use for studying sequences of minimization problems. A key advantage of this approach is that it allows us to deal with infinite dimensional and highly oscillatory functionals.
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