  In real-world, many 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, especially in the field of computer vision.
Recently, the alignment of geometric patterns in high dimension finds several
novel applications, and has attracted more and more attentions. However, the
research is still rather limited in terms of algorithms. To the best of our
knowledge, most existing approaches for high dimensional alignment are just
simple extensions of their counterparts for 2D and 3D cases, and often suffer
from the issues such as high complexities. In this paper, we propose an
effective framework to compress the high dimensional geometric patterns and
approximately preserve the alignment quality. As a consequence, existing
alignment approach can be applied to the compressed geometric patterns and thus
the time complexity is significantly reduced. Our idea is inspired by the
observation that high dimensional data often has a low intrinsic dimension. We
adopt the widely used notion "doubling dimension" to measure the extents of our
compression and the resulting approximation. Finally, we test our method on
both random and real datasets, the 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 running times
(including the times cost for compression) are substantially lower.
