NeuroPMD: Neural Fields for Density Estimation on Product Manifolds
This addresses density estimation challenges in high-dimensional product manifold domains, such as brain connectivity data, with an incremental improvement over existing neural network methods.
The authors tackled density estimation on product Riemannian manifolds by proposing a neural network method that directly parameterizes the density function, trained with penalized maximum likelihood using manifold differential operators. Their approach effectively mitigates the curse of dimensionality and convergence issues, showing clear advantages in simulations and a brain connectivity application.
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.