Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are $E(3)$ equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are equivariant with respect to a single global $E(3)$ transformation, to accommodate inputs made of multiple parts, each of which exhibits local $E(3)$ symmetry.In practical settings, however, the partitioning into individually transforming regions is unknown a priori.Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-$E(3)$ equivariant point networks. Our framework offers an adaptable design to guaranteed bounds on the resulting piecewise $E(3)$ equivariance approximation errors.Our primary insight is that functions which are equivariant with respect to a finer partition (compared to the unknown true partition) will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a compositional design for a partition prediction model. It initiates with a fine partition and incrementally transitions towards a coarser subpartition of the true one, consistently maintaining piecewise equivariance in relation to the current partition.As a result, the equivariance approximation error can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one.We demonstrate the practical effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise $E(3)$ symmetry into network design, showing a distinct improvement in generalization over prior works in terms of generalization accuracy for both classification and segmentation tasks.
Approximately Piecewise E(3) Equivariant Point Networks
Matan Atzmon Francis Williams Jiahui Huang Or Litany