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 H\"older classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded H\"older-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.
翻译:我们研究了以测量运输为基础的非对称密度估计值在海灵根和相关距离上的趋同性。这些估计值代表着以在运输图下选择的参考分布推向所关注的尺度,在这种分布图中,通过一个最大可能性目标(相当于最大限度地减少实验性Kullback-Lebeller损失)或惩罚性版本选择地图。我们通过将基于运输密度表示值的测算法与分析性能相结合,为一般类别的受处罚测量值确定集中性差。然后,我们展示了我们理论对三角Knothe-Rosenblatt(KR)运输在美元维度单位立方体中的影响,并表明,受处罚和未受约束的这种估计值两种版本都实现了比“H”older 类密度最小化的最佳趋同率。具体地说,我们为非受约束性非对受约束的H\“older”型球最高可能性估计率,然后为某些Soboleviled 估测算器和Sieviledwalet 估测器的最佳比率。