# NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

arXiv: https://arxiv.org/abs/2501.02994 · HTML: https://mlsanity.com/paper/2501.02994/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 6 Jan 2025.

## Builds on (most cited first, 9 of 9)

- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Nonparametric Density Estimation for High-Dimensional Data - Algorithms and Applications](https://mlsanity.com/paper/1904.00176/) (2019)
- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families](https://mlsanity.com/paper/2402.09608/) (2024)
- [Multi-dimensional Functional Principal Component Analysis](https://mlsanity.com/paper/1510.04439/) (2015)
- [Directional Statistics and Filtering Using libDirectional](https://mlsanity.com/paper/1712.09718/) (2017)
- [Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems](https://mlsanity.com/paper/2107.14728/) (2021)
- [Scaled torus principal component analysis](https://mlsanity.com/paper/2110.04758/) (2021)

## Built on by (most cited first, 1 of 1)

- [Structured Nonparametric Variational Inference for Dependent Latent Modeling](https://mlsanity.com/paper/2606.15458/) (2026)

## Abstract (arXiv)

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.