NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

TL;DR

NeuroPMD introduces a deep neural network approach for density estimation on high-dimensional product Riemannian manifolds, effectively addressing the curse of dimensionality and convergence issues, with demonstrated advantages in simulations and brain data.

Contribution

It presents a novel neural network architecture and training framework specifically designed for density estimation on complex manifold domains, overcoming limitations of traditional methods.

Findings

Outperforms traditional kernel and basis methods in high dimensions

Effectively mitigates curse of dimensionality

Shows superior results in brain connectivity data analysis

Abstract

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.