2018/10/13 by Jisu Kim, Kim, Jisu, Jaehyeok Shin +5 · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1810.05935
We derive concentration inequalities for the supremum norm of the difference\nbetween a kernel density estimator (KDE) and its point-wise expectation that\nhold uniformly over the selection of the bandwidth and under weaker conditions\non the kernel and the data generating distribution than previously used in the\nliterature. We first propose a novel concept, called the volume dimension, to\nmeasure the intrinsic dimension of the support of a probability distribution\nbased on the rates of decay of the probability of vanishing Euclidean balls.\nOur bounds depend on the volume dimension and generalize the existing bounds\nderived in the literature. In particular, when the data-generating distribution\nhas a bounded Lebesgue density or is supported on a sufficiently well-behaved\nlower-dimensional manifold, our bound recovers the same convergence rate\ndepending on the intrinsic dimension of the support as ones known in the\nliterature. At the same time, our results apply to more general cases, such as\nthe ones of distribution with unbounded densities or supported on a mixture of\nmanifolds with different dimensions. Analogous bounds are derived for the\nderivative of the KDE, of any order. Our results are generally applicable but\nare especially useful for problems in geometric inference and topological data\nanalysis, including level set estimation, density-based clustering, modal\nclustering and mode hunting, ridge estimation and persistent homology.\n