Robust Density Estimation under Besov IPM Losses

Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses

We study the problem of estimating a nonparametric probability density under a large family of losses called Besov IPMs, which include, for example, $\mathcal{L}^p$ distances, total variation distance, and generalizations of both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide both lower and upper bounds, identifying precisely how the choice of loss function and assumptions on the data interact to determine the minimax optimal convergence rate. We also show that linear distribution estimates, such as the empirical distribution or kernel density estimato…

Besov空间上的极小极大自适应在线非参数回归

Minimax Adaptive Online Nonparametric Regression over Besov Spaces

本文针对Besov空间中的在线对抗回归问题，提出无需预知光滑性参数的小波自适应算法，并进一步设计局部自适应扩展以动态追踪空间非均匀光滑性，从而在异质环境中获得超越全局方法的更精细极小极大遗憾界。

Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme

A compound Poisson process whose jump measure and intensity are unknown is observed at finitely many equispaced times. We construct a purely data-driven estimator of the Lévy density $ν$ through the spectral approach using general Calderon--Zygmund integral operators, which include convolution and projection kernels. Assuming minimal tail assumptions, it is shown to estimate $ν$ at the minimax rate of estimation over Besov balls under the losses $L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime (high- and low-frequency). To achieve adaptation in a minimax sense, we us…

On Adaptive Confidence Sets for the Wasserstein Distances

In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the $d-$dimensional torus $\mathbb{T}^d$, in which case $1\leq p\leq 2$, and $\mathbb{R}^d$, for which $p=1$. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent $W_p$-minimax estimation rate. Interestingly, it appears that the po…

重尾贝叶斯非参数自适应

Heavy-tailed Bayesian nonparametric adaptation

提出基于重尾级数先验的贝叶斯非参数平滑性自适应策略，无需采样超参数即可在多种设置下实现极小化极大意义上的自适应收缩率，并在Gaussian回归等模型中达到近最优同时自适应。

Nonparametric Regression Estimation Based on Spatially Inhomogeneous Data: Minimax Global Convergence Rates and Adaptivity

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known compactly supported density g with a finite number of well separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic minimax lower bounds for the global risk of an estimator of f and construct adaptive wavelet nonlinear th…