Equivariance has gained strong interest as a desirable network property that inherently ensures robust generalization. However, when dealing with complex systems such as articulated objects or multi-object scenes, effectively capturing inter-part transformations poses a challenge, as it becomes entangled with the overall structure and local transformations. The interdependence of part assignment and per-part group action necessitates a novel equivariance formulation that allows for their co-evolution. In this paper, we present Banana, a Banach fixed-point network for equivariant segmentation with inter-part equivariance by construction. Our key insight is to iteratively solve a fixed-point problem, where point-part assignment labels and per-part SE(3)-equivariance co-evolve simultaneously. We provide theoretical derivations of both per-step equivariance and global convergence, which induces an equivariant final convergent state. Our formulation naturally provides a strict definition of inter-part equivariance that generalizes to unseen inter-part configurations. Through experiments conducted on both articulated objects and multi-object scans, we demonstrate the efficacy of our approach in achieving strong generalization under inter-part transformations, even when confronted with substantial changes in pointcloud geometry and topology.

翻译：摘要：等变性作为一种理想的网络属性，因其能确保鲁棒的泛化能力而备受关注。然而，在处理复杂系统（如关节物体或多对象场景）时，有效捕捉部件间变换面临挑战，因为这类变换与整体结构及局部变换相互交织。部件分配与逐部件群作用的相互依赖性要求一种新颖的等变性公式，以允许两者协同演化。本文提出Banana——一种基于Banach不动点网络的等变分割方法，通过构造实现部件间等变性。我们的核心见解在于迭代求解一个不动点问题，其中点点-部件分配标签与逐部件SE(3)等变性能同时协同演化。我们提供了单步等变性与全局收敛性的理论推导，从而导出等变的最终收敛状态。该公式自然提供了部件间等变性的严格定义，可推广至未见过的部件间配置。通过在关节物体与多对象扫描数据上的实验，我们证明了该方法在部件间变换下（即使面对点云几何与拓扑的重大变化）仍能实现强大的泛化能力。