FA-KPConv: Introducing Euclidean Symmetries to KPConv via Frame Averaging

TL;DR

The paper addresses the lack of guaranteed exact invariance/equivariance to Euclidean motions in KPConv-based point-cloud networks by introducing Frame Averaging (fa). FA-KPConv wraps any KPConv backbone with a learned-free frame-averaging operator, achieving exact $SE(3)$ invariance/equivariance without adding learnable parameters; the cost scales with the frame size $|\mathcal{F}(\mathbf{X})|$. The authors validate FA-KPConv on 3D shape classification (ModelNet40) and point-cloud registration (3DMatch/3DLoMatch), showing improved robustness to rotations and better data efficiency, particularly in low-data regimes and under challenging test conditions. They also discuss trade-offs, noting that the frame-based prior increases computation and memory by the frame size but does not change the model capacity. Overall, FA-KPConv provides a practical, plug-in mechanism to inject geometric priors into KPConv-based networks with exact symmetry guarantees, enhancing performance in rotation-heavy and data-scarce scenarios.

Abstract

We present Frame-Averaging Kernel-Point Convolution (FA-KPConv), a neural network architecture built on top of the well-known KPConv, a widely adopted backbone for 3D point cloud analysis. Even though invariance and/or equivariance to Euclidean transformations are required for many common tasks, KPConv-based networks can only approximately achieve such properties when training on large datasets or with significant data augmentations. Using Frame Averaging, we allow to flexibly customize point cloud neural networks built with KPConv layers, by making them exactly invariant and/or equivariant to translations, rotations and/or reflections of the input point clouds. By simply wrapping around an existing KPConv-based network, FA-KPConv embeds geometrical prior knowledge into it while preserving the number of learnable parameters and not compromising any input information. We showcase the benefit of such an introduced bias for point cloud classification and point cloud registration, especially in challenging cases such as scarce training data or randomly rotated test data.

Figures (2)

• Figure 1: Diagrams illustrating how (a) the equivariant function $\hat{f}(\textbf{X},\textbf{F})$ and (b) the invariant function $\bar{f}(\textbf{X},\textbf{F})$ can be obtained from $f(\textbf{X},\textbf{F})$ via frame averaging, which is key for defining FA-KPConv. We use small circles to denote the action of a certain group element $g$, and denote $|\mathcal{F}(\textbf{X})|$ by $N$.
• Figure 2: Qualitative comparison of registration results on rotated 3DLoMatch with models trained on 1k samples.