NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

• DOI: 10.48550/arxiv.2501.02994
• Publication Date: 2025-01-01
• Authors: Consagra, William; Gu, Zhiling; Zhang, Zhengwu
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Statistics - Machine Learning; Machine Learning (stat.ML); Machine Learning (cs.LG)

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.