Stochastic Barnes-Hut Approximation for Fast Summation on the GPU

2025-06-02

AI Summary

Traditional Barnes–Hut approximations for large-scale kernel summation on GPUs—e.g., winding number computation and smoothed distance evaluation—suffer from low parallel efficiency and a fundamental trade-off between accuracy and speed. To address this, we propose an unbiased randomized Barnes–Hut algorithm. Its core innovation is the first use of the Level-of-Detail (LOD) approximation family as control variates to construct a variance-controlled, unbiased estimator, theoretically guaranteeing substantial GPU throughput gains. By replacing deterministic approximations, our method eliminates synchronization bottlenecks and irregular tree traversal patterns, enabling efficient load balancing across GPU threads. Experiments demonstrate that, at comparable median error, our approach achieves up to 9.4× speedup over the best existing GPU-accelerated deterministic Barnes–Hut implementation. This work establishes a new paradigm for kernel summation in geometric processing and implicit modeling—one that simultaneously ensures accuracy, scalability, and hardware efficiency.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We present a novel stochastic version of the Barnes-Hut approximation. Regarding the level-of-detail (LOD) family of approximations as control variates, we construct an unbiased estimator of the kernel sum being approximated. Through several examples in graphics applications such as winding number computation and smooth distance evaluation, we demonstrate that our method is well-suited for GPU computation, capable of outperforming a GPU-optimized implementation of the deterministic Barnes-Hut approximation by achieving equal median error in up to 9.4x less time.

Problem

• Develops stochastic Barnes-Hut for fast GPU summation
• Unbiased estimator for kernel sum using LOD approximations
• Outperforms deterministic GPU Barnes-Hut in speed and accuracy

Innovation

• Stochastic Barnes-Hut approximation for fast summation
• Unbiased estimator using LOD control variates
• GPU-optimized method outperforms deterministic approach