解锁测地线Gromov-Wasserstein距离用于3D建模
Unlocking Geodesic Gromov-Wasserstein Distances for 3D Modeling
AI总结：
本文提出EGGroW方法，利用熵正则化Sinkhorn和随机特征高效计算测地线Gromov-Wasserstein距离，应用于3D姿态估计与模板检测，在欧氏方法失效时提供准确且轻量计算的结果。
AI中文摘要：
Gromov-Wasserstein距离（GWDs）通过应用最优传输理论的技术，提供了比较定义在不同度量空间上的概率分布之间的定量方法。因此，GWD在从图匹配问题到3D物体检测的广泛应用中具有潜在价值。然而，其在大规模实际使用中受到显著限制，因为涉及稠密空间内距离矩阵的计算具有立方时间复杂度。尽管在欧几里得度量空间中已提出多种技术（例如涉及可扩展核方法）来解决这一问题，但据我们所知，针对流形上一般测地线距离或其在离散化变体中的图上最短路径距离的类似技术尚未被开发。在本文中，我们提出了高效测地线Gromov-Wasserstein方法（EGGroW），这是一类新的高效算法，旨在利用最近引入的GenusSink方法和随机特征理论，通过熵正则化的Sinkhorn类方法计算测地线Gromov-Wasserstein距离。我们提供了重要的下游应用，即：3D姿态估计和3D模板检测。在后一种设置中，我们将部分3D模板恢复表述为一个分阶段问题：容量约束的场景选择之后是半松弛的模板可见性和对应关系恢复。我们的实证结果表明，当基于欧几里得的标准技术失败时，EGGroW提供准确的解决方案，并且具有轻量计算足迹，正如我们的理论分析所预测的那样。
英文摘要：
\textit{Gromov-Wasserstein Distances} (GWDs) provide quantitative ways of comparing probabilistic distributions defined on different metric spaces by applying techniques from the optimal transport theory. As such, GWD can be potentially useful in a large variety of applications ranging from graph matching problems to 3D object detection. However its practical use at scale is significantly limited by cubic time complexity computations involving dense intra-space distance matrices. Even though in the Euclidean metric spaces several techniques (e.g. involving scalable kernel methods) were proposed to address it, to the best of our knowledge, analogous techniques for general geodesic distances on manifolds, or shortest-path distance on graphs in their discretized variants, were not developed. In this paper, we present \textbf{E}fficient \textbf{G}eodesic \textbf{Gro}mov-\textbf{W}asserstein methods (EGGroW), a new class of efficient algorithms designed to calculate geodesic Gromov-Wasserstein distances with entropic Sinkhorn-like approaches, leveraging recently introduced \textit{GenusSink} methods \citep{genussink} and the theory of random features. We provide important downstream applications, namely: 3D pose estimation and 3D template detection. In the latter setting, we formulate a partial 3D template recovery as a staged problem: capacity-constrained scene selection is followed by semi-relaxed recovery of template visibility and correspondence. Our empirical findings show that EGGroW provides accurate solutions when standard Euclidean-based techniques fail and is characterized by light computational footprint, as our theoretical analysis predicts.
