# CAKE: Convex Adaptive Kernel Density Estimation

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Ravi Sastry Ganti Mahapatruni,
Alexander Gray
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Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, PMLR 15:498-506, 2011.

#### Abstract

In this paper we present a generalization of kernel density estimation called Convex Adaptive Kernel Density Estimation (CAKE) that replaces single bandwidth se- lection by a convex aggregation of kernels at all scales, where the convex aggregation is allowed to vary from one training point to another, treating the fundamental problem of heterogeneous smoothness in a novel way. Learning the CAKE estimator given a training set reduces to solving a single con- vex quadratic programming problem. We derive rates of convergence of CAKE like estimator to the true underlying density under smoothness assumptions on the class and show that given a sufficiently large sample the mean squared error of such estimators is optimal in a minimax sense. We also give a risk bound of the CAKE estimator in terms of its empirical risk. We empirically compare CAKE to other density estimators proposed in the statistics literature for handling heterogeneous smoothness on different synthetic and natural distributions. [pdf][supplementary]

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