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How can we tell complex point clouds with different small scalecharacteristics apart, while disregarding global features? Can we find asuitable transformation of such data in a way that allows to discriminatebetween differences in this sense with statistical guarantees? In this paper,we consider the analysis and classification of complex point clouds as they areobtained, e.g., via single molecule localization microscopy. We focus on thetask of identifying differences between noisy point clouds based on small scalecharacteristics, while disregarding large scale information such as overallsize. We propose an approach based on a transformation of the data via theso-called Distance-to-Measure (DTM) function, a transformation which is basedon the average of nearest neighbor distances. For each data set, we estimatethe probability density of average local distances of all data points and usethe estimated densities for classification. While the applicability isimmediate and the practical performance of the proposed methodology is verygood, the theoretical study of the density estimators is quite challenging, asthey are based on i.i.d. observations that have been obtained via a complicatedtransformation. In fact, the transformed data are stochastically dependent in anon-local way that is not captured by commonly considered dependence measures.Nonetheless, we show that the asymptotic behaviour of the density estimator isdriven by a kernel density estimator of certain i.i.d. random variables byusing theoretical properties of U-statistics, which allows to handle thedependencies via a Hoeffding decomposition. We show via a numerical study andin an application to simulated single molecule localization microscopy data ofchromatin fibers that unsupervised classification tasks based on estimatedDTM-densities achieve excellent separation results.

From Small Scales to Large Scales: Distance-to-Measure Density based Geometric Analysis of Complex Data