论文标题
超越平滑度：将低级分析纳入非参数密度估计
Beyond Smoothness: Incorporating Low-Rank Analysis into Nonparametric Density Estimation
论文作者
论文摘要
最流行的普遍一致的非参数密度估计器的构建和理论分析在一个功能属性上取决于：平滑度。在本文中，我们研究了将多视图潜在变量模型（一种低级模型）纳入非参数密度估计的理论含义。为此，我们对整合多视图模型的直方图估计器进行了广泛的分析。我们的分析最终表明存在一个普遍一致的直方图估计器，该估计器将收敛到任何具有有限数量Lipschitz连续组件的多视图模型，其速率为$ \ widetilde {o}（o}（1/\ sqrt [3] {n} {n}）$ l^1 $ lir^in $ l^1 $错误。相比之下，标准直方图估计器的收敛速度速度慢于$ 1/\ sqrt [d] {n} $在同一类密度上。我们还基于塔克分解引入了一个新的非参数潜在变量模型。我们的估计量的基本实现在实验上表明了对标准直方图估计器的性能改善。我们还提供了基于塔克分解模型的样本复杂性和各种其他结果的彻底分析。因此，我们的论文为将低级技术扩展到非参数设置提供了坚实的理论基础
The construction and theoretical analysis of the most popular universally consistent nonparametric density estimators hinge on one functional property: smoothness. In this paper we investigate the theoretical implications of incorporating a multi-view latent variable model, a type of low-rank model, into nonparametric density estimation. To do this we perform extensive analysis on histogram-style estimators that integrate a multi-view model. Our analysis culminates in showing that there exists a universally consistent histogram-style estimator that converges to any multi-view model with a finite number of Lipschitz continuous components at a rate of $\widetilde{O}(1/\sqrt[3]{n})$ in $L^1$ error. In contrast, the standard histogram estimator can converge at a rate slower than $1/\sqrt[d]{n}$ on the same class of densities. We also introduce a new nonparametric latent variable model based on the Tucker decomposition. A rudimentary implementation of our estimators experimentally demonstrates a considerable performance improvement over the standard histogram estimator. We also provide a thorough analysis of the sample complexity of our Tucker decomposition-based model and a variety of other results. Thus, our paper provides solid theoretical foundations for extending low-rank techniques to the nonparametric setting
