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3D Gaussian Splatting as Markov Chain Monte Carlo
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Reformulates 3D Gaussian Splatting training as MCMC sampling via SGLD, replacing densification/pruning heuristics with a probability-preserving relocation move and an opacity/scale L1 regularizer.
Problem
3D Gaussian Splatting relies on hand-engineered cloning, splitting, pruning and opacity-reset heuristics for placing Gaussians, which makes results depend on good initialization and makes the Gaussian count hard to control.
Approach
Treat the set of Gaussians as MCMC samples from a distribution proportional to exp(-loss); the 3DGS update becomes SGLD once position noise (scaled by covariance and a sigmoid of opacity) is added, and densification/pruning become deterministic relocation moves of dead Gaussians onto live ones that approximately preserve sample probability (opacity and covariance chosen by a sliced-integral matchi
Key results
- On Mip-NeRF 360 with random initialization, the method reaches 29.72 PSNR / 0.89 SSIM / 0.19 LPIPS versus 27.89 / 0.84 / 0.26 for 3DGS with the same number of Gaussians.
- With SfM initialization on Mip-NeRF 360 the method reaches 29.89 PSNR versus 29.30 for 3DGS; on Tanks and Temples it reaches 24.29 versus 23.67.
- The method's random-initialization results are close to its SfM results, showing reduced dependence on initialization.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace 3DGS heuristics with MCMC/SGLD and relocation, using random initialization | 3D Gaussian Splatting · MipNeRF 360 average | PSNR: 27.89 -> 29.72 | Helped |
| replace 3DGS heuristics with MCMC/SGLD and relocation, using random initialization | 3D Gaussian Splatting · Tank & Temples average | PSNR: 21.93 -> 24.21 | Helped |
| replace 3DGS heuristics with MCMC/SGLD and relocation, using random initialization | 3D Gaussian Splatting · OMMO average | PSNR: 28.24 -> 29.31 | Helped |
| replace 3DGS heuristics with MCMC/SGLD and relocation, using random initialization | 3D Gaussian Splatting · NeRF Synthetic average | PSNR: 33.42 -> 33.80 | Helped |
| replace 3DGS heuristics with MCMC/SGLD and relocation, using random initialization | 3D Gaussian Splatting · Deep Blending average | PSNR: 29.55 -> 29.71 | Helped |
| add opacity and scale regularizers (ablation) | Ours · MipNeRF 360 average | PSNR: 23.90 -> 29.72 | Helped |
| add SGLD noise (ablation) | Ours · MipNeRF 360 average | PSNR: 27.41 -> 29.72 | Helped |
| restrict noise to Gaussian locations rather than all parameters (ablation) | Ours · MipNeRF 360 average | PSNR: 29.11 -> 29.72 | Helped |
| add opacity and scale regularizers (ablation) | 3DGS · MipNeRF 360 average | PSNR: 27.89 -> 23.84 | Hurt |
| remove covariance term from location-noise schedule (ablation) | Ours · Tank & Temples | PSNR: 24.21 -> 23.16 | Hurt |
| remove opacity term from location-noise schedule (ablation) | Ours · Tank & Temples | PSNR: 24.21 -> 22.47 | Hurt |
| use a linear rather than exponential noise scheduler (ablation) | Ours · Tank & Temples | PSNR: 24.21 -> 17.64 | Hurt |
Surprising findings
- Adding noise to opacity, scale, and color parameters slightly harmed performance, so noise was applied only to Gaussian locations.
- Ablation shows that the L1 regularizer on the original 3DGS loss hurts classical 3DGS (23.84 PSNR vs 27.89), since it is incompatible with its heuristics.
Limitations
- Relocation preserves probability only approximately, so moves are applied every 100 iterations.
- Inherits the modelling capacity limitations of 3DGS (aliasing, reflections).
- Error-based densification from concurrent work is not incorporated.
Experimental setup
Datasets: NeRF Synthetic Mip-NeRF 360 Tanks and Temples Deep Blending OMMO
Benchmarks: NeRF Synthetic Mip-NeRF 360 Tanks and Temples Deep Blending OMMO
Metrics: PSNR SSIM LPIPS
Baselines: 3DGS (random init) 3DGS (SfM init) NeRF Plenoxels INGP-Big Mip-NeRF Mip-NeRF 360 Bulo et al. error-based densification
Builds on: 3D Gaussian Splatting Stochastic Gradient Langevin Dynamics Adam Mip-NeRF 360
Closest papers by meaning
- Metropolis-Hastings Sampling for 3D Gaussian Reconstruction (NeurIPS 2025 · similarity 0.92)
- GaussianPro: 3D Gaussian Splatting with Progressive Propagation (ICML 2024 · similarity 0.91)
- Fully Explicit Dynamic Gaussian Splatting (NeurIPS 2024 · similarity 0.91)
- GaussianSpa: An "Optimizing-Sparsifying" Simplification Framework for Compact and High-Quality 3D Gaussian Splatting (CVPR 2025 · similarity 0.91)
- DiffGS: Functional Gaussian Splatting Diffusion (NeurIPS 2024 · similarity 0.91)
- Structure Consistent Gaussian Splatting with Matching Prior for Few-shot Novel View Synthesis (NeurIPS 2024 · similarity 0.91)
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