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

TL;DR

Banana introduces a novel fixed-point network that enforces inter-part equivariance for pointcloud segmentation, enabling robust generalization across complex articulated and multi-object scenes by co-evolving part assignments and transformations.

Contribution

The paper proposes Banana, a fixed-point network that naturally models inter-part equivariance, with theoretical guarantees and improved generalization in complex 3D segmentation tasks.

Findings

Achieves strong generalization across unseen inter-part configurations.

Effectively handles complex articulated objects and multi-object scans.

Provides theoretical guarantees of convergence and equivariance.

Abstract

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.