HIGH-DIMENSIONAL NONPARAMETRIC DENSITY ESTIMATION VIA SYMMETRY AND SHAPE CONSTRAINTS

• 成果类型: Article
• 署名作者: Xu, Min; Samworth, Richard J.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1972
• 发表日期: 2021
• 页码: 650-672

摘要

We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on R-p 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 K subset of R-p). 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 n(-4/5), 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.