Learning Coordinate-based Convolutional Kernels for Continuous SE(3) Equivariant and Efficient Point Cloud Analysis

Technical Deep Dive: Learning Coordinate-based Convolutional Kernels for Continuous SE(3) Equivariant and Efficient Point Cloud Analysis

Overview

This work proposes ECKConv, a continuous SE(3) equivariant convolutional network for efficient point cloud analysis. ECKConv adopts the intertwiner framework to embed SE(3) symmetry in the kernel domain and utilizes coordinate-based networks to compute the explicit kernel, enhancing local feature learning and memory scalability.

Problem & Context

• Modern deep learning on point clouds has relied on data augmentation to induce transformation robustness, which is inefficient compared to models retaining symmetry in their architecture.
• Group convolution approaches have faced a trade-off between strict equivariance and model scalability:
  • Discrete group methods can achieve scalability but have discrepancies with the continuity of real-world group spaces.
  • Steerable convolution methods guarantee continuous group equivariance but require expensive computations.
• Previous works based on the intertwiner framework have not presented a continuous SE(3) equivariant method that is also scalable enough to cover large-scale 3D problems.

Methodology

Continuous SE(3) Equivariant Convolution

• ECKConv configures the kernel domain as the double coset space SO(2)\SE(3)/SO(2), enabling continuous SE(3) equivariance.
• The double coset parameters [β, r, z] uniquely represent the SE(3) invariant orbits.

Explicit Kernel using Coordinate-based Networks

• ECKConv computes the explicit kernel by mapping the double coset parameters to coefficients for learnable anchor bases using coordinate-based networks.
• This explicit kernel formulation reduces the memory cost during backpropagation compared to implicit kernel methods.

Architecture

• The ECKConv block performs convolution on centroids by gathering and processing features from the local neighborhood.
• ECKConv is integrated into a U-Net-style residual architecture for end-to-end point cloud tasks.

Data & Experimental Setup

ECKConv is evaluated on the following tasks and datasets:

• Classification and pose registration on ModelNet40
• Part segmentation on ShapeNet
• Indoor semantic segmentation on S3DIS

Baselines include non-equivariant, model-agnostic, and group convolution methods.

Results

• ECKConv achieves the best classification accuracy on ModelNet40, including when the input is randomly rotated.
• ECKConv outperforms baselines in the pose registration task, demonstrating its rigorous SE(3) equivariance.
• ECKConv shows superior part segmentation performance on ShapeNet and state-of-the-art semantic segmentation results on S3DIS, validating its capability to learn local geometries.
• The explicit kernel formulation in ECKConv improves memory efficiency compared to implicit kernel methods.

Limitations

• ECKConv's kernel is isotropic to SE(3) actions, which may limit its expressivity for certain types of point cloud tasks that require anisotropic point features.
• Expanding the resolvable domain of ECKConv while maintaining scalability is an important future research direction.

What Comes Next

Future work could explore ways to incorporate more expressive anisotropic kernels into ECKConv's scalable architecture, balancing the trade-off between equivariance and expressivity.