Recent works have demonstrated that the convergence rate of a nonparametric density estimator can be greatly improved by using a low-rank estimator when the target density is a convex combination of separable probability densities with Lipschitz continuous marginals, i.e. a multiview model. However, this assumption is very restrictive and it is not clear to what degree these findings can be extended to general pdfs. This work answers this question by introducing a new way of characterizing a pdf's complexity, the non-negative Lipschitz spectrum (NL-spectrum), which, unlike smoothness properties, can be used to characterize virtually any pdf. Finite sample bounds are presented that are dependent on the target density's NL-spectrum. From this dimension-independent rates of convergence are derived that characterize when an NL-spectrum allows for a fast rate of convergence.

翻译：近期研究表明，当目标密度函数是可分离概率密度函数的凸组合且其边缘分布满足李普希茨连续条件（即多视角模型）时，通过低秩估计器可大幅提升非参数密度估计器的收敛速度。然而，该假设具有较强局限性，尚不明确这些结论能在多大程度上推广至一般概率密度函数。本研究通过引入一种新的概率密度函数复杂度表征方法——非负李普希茨谱（NL-spectrum），回答了这一问题。与光滑性特征不同，NL-spectrum几乎适用于任意概率密度函数。本文给出了依赖于目标密度NL-spectrum的有限样本界，并由此导出了与维数无关的收敛速度，揭示了何种NL-spectrum能够实现快速收敛。