# B-Spline CNNs on Lie Groups

arXiv: https://arxiv.org/abs/1909.12057 · HTML: https://mlsanity.com/paper/1909.12057/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Sep 2019.

## Built on by (most cited first, 30 of 97)

- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Graph neural networks for materials science and chemistry](https://mlsanity.com/paper/2208.09481/) (2022)
- [Is Attention Better Than Matrix Decomposition?](https://mlsanity.com/paper/2109.04553/) (2021)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptors](https://mlsanity.com/paper/2109.00182/) (2021)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Commutative Lie Group VAE for Disentanglement Learning](https://mlsanity.com/paper/2106.03375/) (2021)
- [E(2)-Equivariant Vision Transformer](https://mlsanity.com/paper/2306.06722/) (2023)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups](https://mlsanity.com/paper/2110.13059/) (2021)
- [Rotation-Equivariant Deep Learning for Diffusion MRI](https://mlsanity.com/paper/2102.06942/) (2021)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [Dilated convolution with learnable spacings](https://mlsanity.com/paper/2112.03740/) (2021)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)
- [A PAC-Bayesian Generalization Bound for Equivariant Networks](https://mlsanity.com/paper/2210.13150/) (2022)
- [Modelling Long Range Dependencies in ND: From Task-Specific to a General Purpose CNN](https://mlsanity.com/paper/2301.10540/) (2023)

## Abstract (arXiv)

Group convolutional neural networks (G-CNNs) can be used to improve classical CNNs by equipping them with the geometric structure of groups. Central in the success of G-CNNs is the lifting of feature maps to higher dimensional disentangled representations, in which data characteristics are effectively learned, geometric data-augmentations are made obsolete, and predictable behavior under geometric transformations (equivariance) is guaranteed via group theory. Currently, however, the practical implementations of G-CNNs are limited to either discrete groups (that leave the grid intact) or continuous compact groups such as rotations (that enable the use of Fourier theory). In this paper we lift these limitations and propose a modular framework for the design and implementation of G-CNNs for arbitrary Lie groups. In our approach the differential structure of Lie groups is used to expand convolution kernels in a generic basis of B-splines that is defined on the Lie algebra. This leads to a flexible framework that enables localized, atrous, and deformable convolutions in G-CNNs by means of respectively localized, sparse and non-uniform B-spline expansions. The impact and potential of our approach is studied on two benchmark datasets: cancer detection in histopathology slides in which rotation equivariance plays a key role and facial landmark localization in which scale equivariance is important. In both cases, G-CNN architectures outperform their classical 2D counterparts and the added value of atrous and localized group convolutions is studied in detail.