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

本文研究了在非参数密度估计中如何引入多视角潜变量模型，探讨了具有唯一一定的连续 Lipschitz 分量的多视角模型的普适一致直方图估计量存在，提出了基于 Tucker 分解的新的非参数潜变量模型，并在实验中发现相对于标准的直方图估计量，本文提出的估计量表现出了明显的性能提高，为将低秩技术扩展到非参数设置提供了坚实的理论基础。