  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 $E(3)$ equivariant, 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 primary insight is
that functions that are equivariant with respect to a finer partition will also
maintain equivariance in relation to the true partition. Leveraging this
observation, we propose a design where the equivariance approximation error at
each layers 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
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 compared to prior works for both classification and
segmentation tasks.
