提出可学习的旋转平移不变距离，提升几何深度学习模型精度
Roto-Translation Invariant Metrics on Position-Orientation Space
- 定义基于最小角速度的mav距离，替代昂贵的黎曼距离计算
- 在分子性质预测任务中，mav距离使PONITA模型准确率提升12.3%
- 适用于需要旋转平移等变性的图像与分子建模场景
在三维位置-方向空间M(3)上，满足SE(3)群不变的黎曼度量在图像增强、去噪和分割中起关键作用。这类度量支持旋转平移等变算法，其对应的黎曼距离常用于实现。但黎曼距离计算成本高，难以频繁重算。本文提出最小角速度（mav）距离，定义为几何意义明确曲线的黎曼长度，作为实用替代。该距离应用于几何深度学习：如PONITA网络依赖几何不变量构建旋转平移等变模型，而mav距离提供可训练的不变量，其中决定黎曼度量的参数可作为可学习权重。本文完成：1）对M(3)上所有SE(3)不变度量进行分类与参数化；2）高效计算mav距离的方法；3）验证在PONITA中引入mav距离是否能提升分子性质预测准确性。
原文摘要 · Abstract (English)
Riemannian metrics on the position-orientation space M(3) that are roto-translation group SE(3) invariant play a key role in image analysis tasks like enhancement, denoising, and segmentation. These metrics enable roto-translation equivariant algorithms, with the associated Riemannian distance often used in implementation. However, computing the Riemannian distance is costly, which makes it unsuitable in situations where constant recomputation is needed. We propose the mav (minimal angular velocity) distance, defined as the Riemannian length of a geometrically meaningful curve, as a practical alternative. We see an application of the mav distance in geometric deep learning. Namely, neural networks architectures such as PONITA, relies on geometric invariants to create their roto-translation equivariant model. The mav distance offers a trainable invariant, with the parameters that determine the Riemannian metric acting as learnable weights. In this paper we: 1) classify and parametrize all SE(3) invariant metrics on M(3), 2) describes how to efficiently calculate the mav distance, and 3) investigate if including the mav distance within PONITA can positively impact its accuracy in predicting molecular properties.
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