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Geometric Algebra Planes: Convex Implicit Neural Volumes
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Introduces GA-Planes, a family of implicit volume representations built from geometric-algebra line, plane and volume feature grids that can be trained convexly or semiconvexly, with 2D theory and 3D segmentation, video segmentation and radiance field experiments.
Problem
Implicit neural volume representations are typically trained with nonconvex optimization, which is slow and sensitive to initialization and hyperparameters; the paper asks whether volume models can be trained with convex or semiconvex objectives while keeping model size and expressiveness.
Approach
GA-Planes combines any subset of line, plane and volume feature grids (inspired by geometric algebra basis elements) with a feature decoder; features are combined by concatenation rather than multiplication so that a convex MLP decoder with frozen ReLU-style gates, or a fused convex model, can be trained to global optimum for convex objectives, while multiplication is used for nonconvex radiance-f
Key results
- On 3D segmentation with 2D tomographic supervision, convex GA-Planes (concatenation) reaches IoU 0.875 while convex Tri-Planes (addition) reaches 0.681.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GA-Planes with concatenated features instead of Tri-Planes with added plane features | 3D segmentation models · 3D segmentation, 2D tomographic supervision | IOU: 0.681 -> 0.875 | Helped |
| Use GA-Planes with concatenated features instead of Tri-Planes with added plane features | 3D segmentation models · 3D segmentation, 3D Space Carving supervision | IOU: 0.642 -> 0.932 | Helped |
| Use GA-Planes with concatenated features instead of Tri-Planes with added plane features | video segmentation models · Video segmentation, held-out test frames | IOU: 0.557 -> 0.913 | Helped |
| Train GA-Planes with concatenated features using convex instead of nonconvex optimization | GA-Planes · 3D segmentation, 2D tomographic supervision | IOU: 0.880 -> 0.875 | No clear effect |
| Train Tri-Planes with added features using convex instead of nonconvex optimization | Tri-Planes · 3D segmentation, 2D tomographic supervision | IOU: 0.863 -> 0.681 | Hurt |
| Train Tri-Planes with added features using convex instead of nonconvex optimization | Tri-Planes · 3D segmentation, 3D Space Carving supervision | IOU: 0.941 -> 0.642 | Hurt |
| Use feature multiplication instead of concatenation (ablation) | GA-Planes · 3D segmentation, 2D tomographic supervision | IOU: 0.880 -> 0.877 | No clear effect |
| Use feature multiplication instead of concatenation (ablation) | GA-Planes · 3D segmentation, 3D Space Carving supervision | IOU: 0.964 -> 0.926 | Hurt |
| Use feature multiplication instead of concatenation (ablation) | GA-Planes · Video segmentation, held-out test frames | IOU: 0.981 -> 0.974 | Hurt |
Surprising findings
- Nonconvex Tri-Planes with addition performs poorly under convex and semiconvex training on segmentation tasks, while GA-Planes stays stable across all three formulations.
- Feature multiplication gives no clear advantage over concatenation for nonlinear MLP decoders in the video segmentation task.
Limitations
- Focuses on 3D or smaller representations; dynamic volumes are not explored.
- Evaluated only on reconstruction tasks, not generation.
- Uses first-order optimization for all models; convex formulation not tested with dedicated convex solvers.
Experimental setup
Datasets: NeRF-Blender (NeRF synthetic) scenes SciPy astronaut image skateboarding video (segmented with YOLOv8 and SAM)
Benchmarks: NeRF-Blender NeRF-Blender lego opacity masks (3D segmentation) skateboarding video segmentation
Metrics: PSNR SSIM LPIPS IoU
Baselines: Mip-NeRF TensoRF K-Planes Tri-Planes Plenoxels DVGO
Builds on: convex neural networks (Pilanci & Ergen) TensoRF K-Planes Tri-Planes Plenoxels NeRF
Closest papers by meaning
- Neural Processing of Tri-Plane Hybrid Neural Fields (ICLR 2024 · similarity 0.85)
- Gaussian Herding across Pens: An Optimal Transport Perspective on Global Gaussian Reduction for 3DGS (NeurIPS 2025 · similarity 0.84)
- Neural Signed Distance Function Inference through Splatting 3D Gaussians Pulled on Zero-Level Set (NeurIPS 2024 · similarity 0.84)
- GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems (NeurIPS 2026 · similarity 0.84)
- Compact 3D Gaussian Representation for Radiance Field (CVPR 2024 · similarity 0.84)
- Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals (NeurIPS 2025 · similarity 0.84)
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