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Abstract:
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In this paper, the lower dimensional topological features occurring as zero density regions of density functions are introduced and thoroughly investigated. We develop a shrinkage scheme for the radii of families of covering balls for a dataset of growing size. As the size of the i.i.d. dataset goes to infinity, when the shrinkage rate of the covering balls is appropriately chosen, the balls that are close to the zero density region do not contain any observed data points, while each ball far away from the zero density region contains at least one observed data point. Our result gives sufficient conditions for such a shrinkage covering ball scheme to exist and work. With an appropriate rate of shrinkage of the radius of these balls as a function of sample size, we can detect lower dimensional zero density regions while guarding against false detections.
We also supplement the theoretic result with computer simulation data, where the union of empty covering balls locates the topological feature. This result is relevant in other branches like non-parametric density estimation, manifold learning, and useful in applications like image segmentation.
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