Many real-world problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns in the field of computer vision. Recently, the alignment problem in high dimensions finds several novel applications in practice. However, the research is still rather limited in the algorithmic aspect. To the best of our knowledge, most existing approaches are just simple extensions of their counterparts for 2D and 3D cases, and often suffer from the issues such as high computational complexities. In this paper, we propose an effective framework to compress the high dimensional geometric patterns. Any existing alignment method can be applied to the compressed geometric patterns and the time complexity can be significantly reduced. Our idea is inspired by the observation that high dimensional data often has a low intrinsic dimension. Our framework is a ``data-dependent'' approach that has the complexity depending on the intrinsic dimension of the input data. Our experimental results reveal that running the alignment algorithm on compressed patterns can achieve similar qualities, comparing with the results on the original patterns, but the runtimes (including the times cost for compression) are substantially lower.

翻译：许多现实世界的问题可以表述为两个几何模式之间的对齐。此前，大量研究聚焦于计算机视觉领域中二维或三维模式的对齐。近年来，高维对齐问题在实际应用中发现了若干新场景，然而算法方面的研究仍然相当有限。据我们所知，现有大多数方法仅是二维和三维情形下对应方法的简单扩展，且常面临高计算复杂度等问题。本文提出了一种有效压缩高维几何模式的框架。任何现有对齐方法均可应用于压缩后的几何模式，从而显著降低时间复杂度。我们的灵感源于高维数据通常具有低内在维度的观察。该框架是一种“数据依赖”方法，其复杂度取决于输入数据的内在维度。实验结果表明，相较于原始模式上的对齐结果，在压缩模式上运行对齐算法可达到相近的质量，但运行时间（包括压缩时间）显著降低。