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

Point cloud registration critically depends on local features that are both distinctive and robust to arbitrary 3D rotations. Existing learning-based methods typically approximate rotation invariance via fragile local reference frames or extensive data augmentation, providing only empirical invariance and often degrading under unseen rotational transformations. In this paper, we propose SHReg, a strictly rotation-equivariant point cloud registration framework grounded in the representation theory of $SO(3)$. By representing local geometric features as irreducible representations of $SO(3)$, SHReg guarantees exact equivariance under arbitrary rotations without relying on local reference frames. Built upon a spherical-harmonics-based equivariant backbone, SHReg jointly learns rotation-invariant descriptors for robust correspondence matching and rotation-equivariant features that preserve fine-grained orientation information. The preserved equivariant structure enables each correspondence to directly hypothesize a rigid transformation, reducing reliance on large-scale hypothesis sampling in conventional RANSAC-based pipelines and leading to improved robustness under challenging rotational variations. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI demonstrate that SHReg consistently outperforms state-of-the-art methods in registration accuracy, particularly under large rotational perturbations.

The recent receptance weighted key value (RWKV) model combines RNN-style recurrence, offering a linear-complexity alternative to Transformers' quadratic self-attention for modeling global dependencies. However, when directly applied to point clouds, RWKV, originally developed for sequential text, struggles to capture local geometric structures and model spatial dependencies effectively. To address this, we propose the P-RWKV block, which bridges the gap between sequence modeling and irregular 3D geometry while preserving the efficiency advantages of RWKV. It consists of a Local Perception Expansion (LPE) component to expand contextual perception along the spatio-temporal sequence and a Spatial Context Enhancement (SCE) component to strengthen spatial awareness. To validate the effectiveness of P-RWKV for point cloud understanding, we construct PointER, a single-modality self-supervised representation learning framework whose encoder is composed of stacked P-RWKV blocks. Furthermore, we extend P-RWKV to a cross-modality setting and integrate the proposed core sub-modules into multiple architectures, demonstrating strong plug-and-play flexibility and architectural generality. Extensive experiments show that the P-RWKV block and its key sub-modules achieve competitive performance across various tasks with lower computational cost and inference latency. Code will be released upon acceptance.

We present an alternative way of solving the steerable kernel constraint that appears in the design of steerable equivariant convolutional neural networks. We find explicit real and complex bases which are ready to use, for different symmetry groups and for feature maps of arbitrary tensor type. A major advantage of this method is that it bypasses the need to numerically or analytically compute Clebsch-Gordan coefficients and works directly with the representations of the input and output feature maps. The strategy is to find a basis of kernels that respect a simpler invariance condition at some point $x_0$, and then steer it with the defining equation of steerability to move to some arbitrary point $x=g\cdot x_0$. This idea has already been mentioned in the literature before, but not advanced in depth and with some generality. Here we describe how it works with minimal technical tools to make it accessible for a general audience.