# Geometric Algebra Planes: Convex Implicit Neural Volumes

arXiv: https://arxiv.org/abs/2411.13525 · HTML: https://mlsanity.com/paper/2411.13525/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 20 Nov 2024.

## Builds on (most cited first, 26 of 26)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [Efficient Geometry-aware 3D Generative Adversarial Networks](https://mlsanity.com/paper/2112.07945/) (2021)
- [Direct Voxel Grid Optimization: Super-fast Convergence for Radiance Fields Reconstruction](https://mlsanity.com/paper/2111.11215/) (2021)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [K-Planes: Explicit Radiance Fields in Space, Time, and Appearance](https://mlsanity.com/paper/2301.10241/) (2023)
- [Nerfstudio: A Modular Framework for Neural Radiance Field Development](https://mlsanity.com/paper/2302.04264/) (2023)
- [DeepVoxels: Learning Persistent 3D Feature Embeddings](https://mlsanity.com/paper/1812.01024/) (2018)
- [Advances in Neural Rendering](https://mlsanity.com/paper/2111.05849/) (2021)
- [Mixture of Volumetric Primitives for Efficient Neural Rendering](https://mlsanity.com/paper/2103.01954/) (2021)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [MERF: Memory-Efficient Radiance Fields for Real-time View Synthesis in Unbounded Scenes](https://mlsanity.com/paper/2302.12249/) (2023)
- [Neural Networks are Convex Regularizers: Exact Polynomial-time Convex Optimization Formulations for Two-layer Networks](https://mlsanity.com/paper/2002.10553/) (2020)
- [SCADE: NeRFs from Space Carving with Ambiguity-Aware Depth Estimates](https://mlsanity.com/paper/2303.13582/) (2023)
- [A Comprehensive Survey on 3D Content Generation](https://mlsanity.com/paper/2402.01166/) (2024)
- [Path Regularization: A Convexity and Sparsity Inducing Regularization for Parallel ReLU Networks](https://mlsanity.com/paper/2110.09548/) (2021)
- [Benchmarking Neural Radiance Fields for Autonomous Robots: An Overview](https://mlsanity.com/paper/2405.05526/) (2024)
- [Neural radiance fields in the industrial and robotics domain: applications, research opportunities and use cases](https://mlsanity.com/paper/2308.07118/) (2023)
- [A Review of 3D Reconstruction Techniques for Deformable Tissues in Robotic Surgery](https://mlsanity.com/paper/2408.04426/) (2024)

## Built on by (most cited first, 5 of 5)

- [Optimizing Rank for High-Fidelity Implicit Neural Representations](https://mlsanity.com/paper/2512.14366/) (2025)
- [Low-Rank Adaptation of Neural Fields](https://mlsanity.com/paper/2504.15933/) (2025)
- [Grids Often Outperform Implicit Neural Representations at Compressing Dense Signals](https://mlsanity.com/paper/2506.11139/) (2025)
- [3D Field of Junctions: A Noise-Robust, Training-Free Structural Prior for Volumetric Inverse Problems](https://mlsanity.com/paper/2603.02149/) (2026)
- [Fast Implicit Neural Light Field Representation via Geometric Decomposition and Multi-Resolution Low-Rank Features](https://mlsanity.com/paper/2608.13949/) (2026)

## Abstract (arXiv)

Volume parameterizations abound in recent literature, from the classic voxel grid to the implicit neural representation and everything in between. While implicit representations have shown impressive capacity and better memory efficiency compared to voxel grids, to date they require training via nonconvex optimization. This nonconvex training process can be slow to converge and sensitive to initialization and hyperparameter choices that affect the final converged result. We introduce a family of models, GA-Planes, that is the first class of implicit neural volume representations that can be trained by convex optimization. GA-Planes models include any combination of features stored in tensor basis elements, followed by a neural feature decoder. They generalize many existing representations and can be adapted for convex, semiconvex, or nonconvex training as needed for different inverse problems. In the 2D setting, we prove that GA-Planes is equivalent to a low-rank plus low-resolution matrix factorization; we show that this approximation outperforms the classic low-rank plus sparse decomposition for fitting a natural image. In 3D, we demonstrate GA-Planes' competitive performance in terms of expressiveness, model size, and optimizability across three volume fitting tasks: radiance field reconstruction, 3D segmentation, and video segmentation.