3DGS$^2$: Near Second-order Converging 3D Gaussian Splatting (2501.13975v2)

Abstract: 3D Gaussian Splatting (3DGS) has emerged as a mainstream solution for novel view synthesis and 3D reconstruction. By explicitly encoding a 3D scene using a collection of Gaussian kernels, 3DGS achieves high-quality rendering with superior efficiency. As a learning-based approach, 3DGS training has been dealt with the standard stochastic gradient descent (SGD) method, which offers at most linear convergence. Consequently, training often requires tens of minutes, even with GPU acceleration. This paper introduces a (near) second-order convergent training algorithm for 3DGS, leveraging its unique properties. Our approach is inspired by two key observations. First, the attributes of a Gaussian kernel contribute independently to the image-space loss, which endorses isolated and local optimization algorithms. We exploit this by splitting the optimization at the level of individual kernel attributes, analytically constructing small-size Newton systems for each parameter group, and efficiently solving these systems on GPU threads. This achieves Newton-like convergence per training image without relying on the global Hessian. Second, kernels exhibit sparse and structured coupling across input images. This property allows us to effectively utilize spatial information to mitigate overshoot during stochastic training. Our method converges an order faster than standard GPU-based 3DGS training, requiring over $10\times$ fewer iterations while maintaining or surpassing the quality of the compared with the SGD-based 3DGS reconstructions.

• The paper presents a localized Newton optimization realizing near second-order convergence for efficient 3D Gaussian Splatting.
• It demonstrates a 10x reduction in iteration count compared to traditional SGD, significantly accelerating training and enhancing quality.
• The approach exploits independent optimization of Gaussian kernel attributes with sparse inter-image coupling to mitigate overshoot and boost stability.

An Examination of Near Second-order Converging 3D Gaussian Splatting

The paper "3DGS\textsuperscript{2}: Near Second-order Converging 3D Gaussian Splatting" introduces a novel approach to enhance the efficiency and efficacy of 3D Gaussian Splatting (3DGS) for novel view synthesis and 3D reconstruction. The work presents a training methodology that embraces nearly a second-order convergence through a localized Newton optimization process, diverging from the standard stochastic gradient descent (SGD) methods traditionally employed in this domain.

Context and Motivation

3D Gaussian Splatting has gained traction as an effective method for representing and rendering 3D scenes by using a collection of Gaussian kernels. This method competes with Neural Radiance Fields (NeRFs) by offering an alternative rendering approach primarily characterized by higher computational efficiency due to its reliance on rasterization rather than ray marching. Conventionally, 3DGS training has involved standard SGD, yielding linear convergence and resulting in relatively long training times even with GPU acceleration.

Methodological Innovations

The authors propose a localized Newton optimization method based on two primary observations:

1. Independent Contribution of Gaussian Kernel Attributes: Each Gaussian kernel's attributes can be optimized independently, facilitating isolated and local optimization. The approach leverages the independence of attributes to split the optimization task into smaller, more manageable Newton systems. These are efficiently solved using GPU threads, approximating Newton-like convergence without the overhead of computing a global Hessian matrix.
2. Sparse and Structured Kernel Coupling Across Input Images: Kernels display sparse coupling across different input images, which allows for the effective utilization of spatial information to adjust for potential overshoot during training iterations. By sampling spatially adjacent viewpoints, the algorithm mitigates the probability of convergence overshoot, enhancing stability.

The paper claims that their method reduces the required iterations by over an order of magnitude compared to SGD-based training while maintaining or even improving the quality of the rendered output.

Strong Numerical Results

The empirical results demonstrated in the paper substantiate the efficiency of the proposed method. The approach achieves considerable reductions in required iterations, demonstrating a $10 \times$ improvement in iteration efficiency over traditional SGD training. Such efficiency is attained without compromising reconstruction quality, as confirmed through standard metrics like SSIM, PSNR, and LPIPS. The substantial reduction in training time from minutes to seconds marks a significant achievement in the field.

Implications and Future Directions

This research has both practical and theoretical implications. Practically, the significant reduction in rendering time facilitates real-time applications of 3DGS in augmented reality and other interactive experiences, where computational efficiency is paramount. Theoretically, the insights into kernel attribute independence and sparse coupling present avenues for further exploration in multi-view optimization and adaptive sampling strategies.

The prospect of integrating this localized Newton optimization framework into other learning-based graphics techniques offers an exciting trajectory for future AI developments. Moreover, exploring hybrid methods that combine the strengths of NeRF's implicit representations with 3DGS's efficiency could yield even more robust solutions for 3D scene reconstruction and synthesis.

In summary, the paper advances the current understanding of optimization in 3D Gaussian Splatting, providing a promising second-order convergent training methodology that significantly enhances training efficiency and quality while operating within the constraints of available computational resources.