Steepest Descent Density Control for Compact 3D Gaussian Splatting

3D Gaussian Splatting (3DGS) has emerged as a powerful technique for real-time, high-resolution novel view synthesis.By representing scenes as a mixture of Gaussian primitives, 3DGS leverages GPU rasterization pipelines for efficient rendering and reconstruction. To optimize scene coverage and capture fine details, 3DGS employs a densification algorithm to generate additional points.However, this process often leads to redundant point clouds, resulting in excessive memory usage, slower performance, and substantial storage demands--posing significant challenges for deployment on resource-constrained devices. To address this limitation, we propose a theoretical framework that demystifies and improves density control in 3DGS. Our analysis reveals that splitting is crucial for escaping saddle points. Through an optimization-theoretic approach, we establish the necessary conditions for densification, determine the minimal number of offspring Gaussians, identify the optimal parameter update direction, and provide an analytical solution for normalizing off-spring opacity. Building on these insights, we introduce **SteepGS**, incorporating *steepest density control*, a principled strategy that minimizes loss while maintaining a compact point cloud. SteepGS achieves a $\sim$ 50\% reduction in Gaussian points without compromising rendering quality, significantly enhancing both efficiency and scalability.