Density estimation is a fundamental task in statistics and machine learning applications. Kernel density estimation is a powerful tool for non-parametric density estimation in low dimensions; however, its performance is poor in higher dimensions. Moreover, its prediction complexity scale linearly with more training data points. This paper presents a method for neural density estimation that can be seen as a type of kernel density estimation, but without the high prediction computational complexity. The method is based on density matrices, a formalism used in quantum mechanics, and adaptive Fourier features. The method can be trained without optimization, but it could be also integrated with deep learning architectures and trained using gradient descent. Thus, it could be seen as a form of neural density estimation method. The method was evaluated in different synthetic and real datasets, and its performance compared against state-of-the-art neural density estimation methods, obtaining competitive results.

翻译：密度估计是统计和机器学习应用的一项基本任务。核心密度估计是低维的非参数密度估计的有力工具;然而,其性能在较高维度方面较差。此外,其预测复杂性线性规模与更多的培训数据点比较。本文提出了神经密度估计方法,可视为内核密度的一种类型,但没有较高的预测计算复杂性。该方法基于密度矩阵、量子力学中使用的正规主义和适应性Fourier特性。该方法可以不优化地加以培训,但也可以与深学习结构相结合,并利用梯度下降进行训练。因此,该方法可被视为神经密度估计方法的一种形式。该方法在不同的合成和真实数据集中进行了评估,其性能与最先进的神经密度估计方法相比较,获得了竞争性的结果。