We propose a novel deep neural network methodology for density estimation on product Riemannian manifold domains. In our approach, the network directly parameterizes the unknown density function and is trained using a penalized maximum likelihood framework, with a penalty term formed using manifold differential operators. The network architecture and estimation algorithm are carefully designed to handle the challenges of high-dimensional product manifold domains, effectively mitigating the curse of dimensionality that limits traditional kernel and basis expansion estimators, as well as overcoming the convergence issues encountered by non-specialized neural network methods. Extensive simulations and a real-world application to brain structural connectivity data highlight the clear advantages of our method over the competing alternatives.

翻译：本文提出了一种新颖的深度神经网络方法，用于产品黎曼流形域上的密度估计。在我们的方法中，网络直接参数化未知密度函数，并通过惩罚极大似然框架进行训练，其中惩罚项利用流形微分算子构建。网络架构与估计算法经过精心设计，以应对高维产品流形域的挑战，有效缓解了限制传统核方法与基展开估计器的维度灾难问题，同时克服了非专用神经网络方法所面临的收敛难题。大量仿真实验及在脑结构连接数据上的实际应用均表明，本方法相较于现有竞争方案具有显著优势。