NeuroPMD: Neural Fields for Density Estimation on Product Manifolds
提出一种深度神经网络方法，直接在乘积黎曼流形上参数化未知密度函数，通过惩罚最大似然框架训练，有效应对高维乘积流形上的维度灾难和收敛问题，在脑结构连接数据等应用中优于传统方法。
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 computational 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.