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Equivariance has gained strong interest as a desirable network property thatinherently ensures robust generalization. However, when dealing with complexsystems such as articulated objects or multi-object scenes, effectivelycapturing inter-part transformations poses a challenge, as it becomes entangledwith the overall structure and local transformations. The interdependence ofpart assignment and per-part group action necessitates a novel equivarianceformulation that allows for their co-evolution. In this paper, we presentBanana, a Banach fixed-point network for equivariant segmentation withinter-part equivariance by construction. Our key insight is to iterativelysolve a fixed-point problem, where point-part assignment labels and per-partSE(3)-equivariance co-evolve simultaneously. We provide theoretical derivationsof both per-step equivariance and global convergence, which induces anequivariant final convergent state. Our formulation naturally provides a strictdefinition of inter-part equivariance that generalizes to unseen inter-partconfigurations. Through experiments conducted on both articulated objects andmulti-object scans, we demonstrate the efficacy of our approach in achievingstrong generalization under inter-part transformations, even when confrontedwith substantial changes in pointcloud geometry and topology.

Banana: Banach Fixed-Point Network for Pointcloud Segmentation with Inter-Part Equivariance