Abstract
We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are K-homothetic (i.e., scalar multiples of a convex body ). When K is known, we prove that the K-homothetic log-concave maximum likelihood estimator based on n independent observations from such a density achieves the minimax optimal rate of convergence with respect to, for example, squared Hellinger loss, of order , independent of p. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst case and adaptive risk bounds in cases where K is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when n and p are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data.
Citation
Min Xu. Richard J. Samworth. "High-dimensional nonparametric density estimation via symmetry and shape constraints." Ann. Statist. 49 (2) 650 - 672, April 2021. https://doi.org/10.1214/20-AOS1972
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