Geometric Algebra Planes: Convex Implicit Neural Volumes

Irmak Sivgin, Sara Fridovich-Keil, Gordon Wetzstein, Mert Pilanci
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:55828-55853, 2025.

Abstract

Volume parameterizations abound in recent literature, encompassing methods from classic voxel grids to implicit neural representations. While implicit representations offer impressive capacity and improved memory efficiency compared to voxel grids, they traditionally require training through nonconvex optimization, which can be slow and sensitive to initialization and hyperparameters. We introduce GA-Planes, a novel family of implicit neural volume representations inspired by Geometric Algebra that can be trained using convex optimization, addressing the limitations of nonconvex methods. GA-Planes models generalize many existing representations including any combination of features stored in tensor basis elements followed by a neural feature decoder, and can be adapted to convex or nonconvex training as needed for various inverse problems. In the 2D setting, we prove GA-Planes models are equivalent to a low-rank plus low-resolution matrix factorization that outperforms the classic low-rank plus sparse decomposition for fitting a natural image. In 3D, GA-Planes models exhibit competitive expressiveness, model size, and optimizability across tasks such as radiance field reconstruction, 3D segmentation, and video segmentation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-sivgin25a, title = {Geometric Algebra Planes: Convex Implicit Neural Volumes}, author = {Sivgin, Irmak and Fridovich-Keil, Sara and Wetzstein, Gordon and Pilanci, Mert}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {55828--55853}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/sivgin25a/sivgin25a.pdf}, url = {https://proceedings.mlr.press/v267/sivgin25a.html}, abstract = {Volume parameterizations abound in recent literature, encompassing methods from classic voxel grids to implicit neural representations. While implicit representations offer impressive capacity and improved memory efficiency compared to voxel grids, they traditionally require training through nonconvex optimization, which can be slow and sensitive to initialization and hyperparameters. We introduce GA-Planes, a novel family of implicit neural volume representations inspired by Geometric Algebra that can be trained using convex optimization, addressing the limitations of nonconvex methods. GA-Planes models generalize many existing representations including any combination of features stored in tensor basis elements followed by a neural feature decoder, and can be adapted to convex or nonconvex training as needed for various inverse problems. In the 2D setting, we prove GA-Planes models are equivalent to a low-rank plus low-resolution matrix factorization that outperforms the classic low-rank plus sparse decomposition for fitting a natural image. In 3D, GA-Planes models exhibit competitive expressiveness, model size, and optimizability across tasks such as radiance field reconstruction, 3D segmentation, and video segmentation.} }
Endnote
%0 Conference Paper %T Geometric Algebra Planes: Convex Implicit Neural Volumes %A Irmak Sivgin %A Sara Fridovich-Keil %A Gordon Wetzstein %A Mert Pilanci %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-sivgin25a %I PMLR %P 55828--55853 %U https://proceedings.mlr.press/v267/sivgin25a.html %V 267 %X Volume parameterizations abound in recent literature, encompassing methods from classic voxel grids to implicit neural representations. While implicit representations offer impressive capacity and improved memory efficiency compared to voxel grids, they traditionally require training through nonconvex optimization, which can be slow and sensitive to initialization and hyperparameters. We introduce GA-Planes, a novel family of implicit neural volume representations inspired by Geometric Algebra that can be trained using convex optimization, addressing the limitations of nonconvex methods. GA-Planes models generalize many existing representations including any combination of features stored in tensor basis elements followed by a neural feature decoder, and can be adapted to convex or nonconvex training as needed for various inverse problems. In the 2D setting, we prove GA-Planes models are equivalent to a low-rank plus low-resolution matrix factorization that outperforms the classic low-rank plus sparse decomposition for fitting a natural image. In 3D, GA-Planes models exhibit competitive expressiveness, model size, and optimizability across tasks such as radiance field reconstruction, 3D segmentation, and video segmentation.
APA
Sivgin, I., Fridovich-Keil, S., Wetzstein, G. & Pilanci, M.. (2025). Geometric Algebra Planes: Convex Implicit Neural Volumes. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:55828-55853 Available from https://proceedings.mlr.press/v267/sivgin25a.html.

Related Material