发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出结构自适应树场积分器（STAD-TFIs），利用树结构分解和二维FFT实现近线性积分，在合成树、网格最优传输和视觉任务中验证了高效性与准确性。
AI 中文摘要
我们提出了一类新的近线性算法，用于高效积分定义在树上、具有距离依赖核的一般张量场，即结构自适应树场积分器（STAD-TFIs）。STAD-TFIs通过围绕路径主干和单顶点分隔符构建的分解来利用树的底层结构，并使用二维快速傅里叶变换来联合计算相互作用。通过利用这种结构信息，STAD-TFIs比其常规的高效树场积分器（TFI）对应方法实现了更高效的计算积分。我们对我们提出的方法进行了详细的理论分析，并辅以详尽的实证评估，范围从合成树上的速度测试，到在真实网格上加速基于Sinkhorn的最优传输算法的松弛，再到用于视觉任务的拓扑注意力变换器。据我们所知，我们提供了一些首批结果，表明通过应用快速TFI方法，可以推导出测地线Sinkhorn最优传输解的高效计算且准确的松弛。
英文摘要
We present a new class of near-linear algorithms for efficiently integrating general tensor fields defined on trees with distance dependent kernels, the Structure-Adaptive Tree Field Integrators (STAD-TFIs). STAD-TFIs exploit the tree's underlying structure through decompositions built around path backbones and single vertex separators, and use two-dimensional fast Fourier transforms to compute interactions jointly. By exploiting this structural information, STAD-TFIs achieve more computationally efficient integration than their regular efficient tree field integrators (TFI) counterparts. We provide a detailed theoretical analysis of our proposed approach and complement it with an exhaustive empirical evaluation, ranging from speed tests on synthetic trees, through accelerated Sinkhorn-based relaxations of the Optimal Transport algorithms on real meshes, to Topological Attention Transformers for vision tasks. To the best of our knowledge, we provide some of the first results showing that efficient to compute and accurate relaxations of the geodesic Sinkhorn-based solutions of the Optimal Transport problem can be derived by applying fast TFI methods.
