April 2021 High-dimensional nonparametric density estimation via symmetry and shape constraints
Min Xu, Richard J. Samworth
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Ann. Statist. 49(2): 650-672 (April 2021). DOI: 10.1214/20-AOS1972


We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on Rp 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 KRp). 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 n4/5, 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.


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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


Received: 1 March 2019; Revised: 1 December 2019; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1972

Primary: 62G07

Keywords: Density estimation , High-dimensional statistics , nonparametric estimation , Shape-constrained estimation

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • April 2021
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