NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

2025-01-06

AI Summary

Probability density estimation on high-dimensional product Riemannian manifolds is hindered by the curse of dimensionality; conventional methods suffer from poor scalability, while generic neural networks lack geometric awareness and exhibit unstable convergence. Method: We propose the first neural-field-based direct density modeling framework for such spaces. Our approach constructs a manifold-adaptive neural field and incorporates Riemannian differential operators—particularly the Laplace–Beltrami operator—as geometrically informed regularizers within a penalized maximum likelihood objective. Contribution/Results: This work pioneers the extension of neural fields to product Riemannian manifolds. By embedding differential-geometric priors, it mitigates the curse of dimensionality and enhances training stability. Experiments on multiple high-dimensional synthetic manifold datasets and real human brain structural connectome data demonstrate significant improvements over kernel density estimation, basis-function expansions, and unstructured neural network baselines.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• High-dimensional Data
• Riemannian Manifolds
• Neural Networks

Innovation

• Deep Neural Network
• Product Riemannian Manifold
• Density Estimation