论文标题
在最小值密度估计通过测量运输估算
On minimax density estimation via measure transport
论文作者
论文摘要
我们研究基于度量传输的非参数密度估计器的收敛性和相关距离。这些估计量代表了利息的度量，作为传输图下选择的参考分布的推动力，其中图是通过最大似然目标选择的地图（等效地，将经验的kullback-leibler损失最小化）或其惩罚版本。我们通过将M估计的技术与基于运输的密度表示的分析性能相结合，为一般惩罚措施估计量的一般类别的措施运输估计器建立了浓度不平等。然后，我们证明了我们理论对三角形诺斯布拉特（KR）在$ d $维单元方面的运输的含义，并表明该估计器的惩罚和未化的版本都可以实现高于Hölder浓度的最小值最佳收敛速率。具体而言，我们为有限的Hölder-type球，然后在某些Sobolev-PenalateAlizatization估计器和筛分的小波估计器中建立了未覆盖的非参数最大似然估计的最佳速率。
We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the transport-based density representation. We then demonstrate the implications of our theory for the case of triangular Knothe-Rosenblatt (KR) transports on the $d$-dimensional unit cube, and show that both penalized and unpenalized versions of such estimators achieve minimax optimal convergence rates over Hölder classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded Hölder-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.
