ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds

Abstract

In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries.We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision. References

Low Dimensional Invariant Embeddings for Universal Geometric Learning

“…is SO(2) invariant and separate orbits. Each one of these linear maps W (j) is parameterized by 2n real numbers, while our invariants in (8) are parameterized by 3n parameters (when d = 2). When d = 2, it would be natural to look for separating invariants of the form (10), where the W (j) are SO(d) equivariant linear maps from R d×n to R d , and avoid the additional determinant term we use in (8).…”

“…Each one of these linear maps W (j) is parameterized by 2n real numbers, while our invariants in (8) are parameterized by 3n parameters (when d = 2). When d = 2, it would be natural to look for separating invariants of the form (10), where the W (j) are SO(d) equivariant linear maps from R d×n to R d , and avoid the additional determinant term we use in (8). However, in Proposition A.1 in the appendix we show that when d = 2, the only linear SO(d) equivariant maps are of the form X → Xw with w ∈ R n .…”

“…Analogously, it is desirable to devise invariant (or equivariant) neural networks which are universal in the sense that they can approximate all continuous invariant (or equivariant) functions. This question has been studied in many works such as [18,44,31,8,16,22,33,27,35].…”

A Simple and Universal Rotation Equivariant Point-cloud Network

“…Thus it is desirable to obtain simpler equivariant networks with universality guarantees. This goal was obtained for the simpler cases of 2D point clouds (Bökman et al, 2021) or 3D point clouds with distinct principal eigenvalues (Puny et al, 2021).…”