A Data-dependent Approach for High Dimensional (Robust) Wasserstein Alignment

TL;DR

This paper introduces a data-dependent framework for high-dimensional geometric pattern alignment that reduces computational complexity by compressing data based on its intrinsic dimension, maintaining quality while improving efficiency.

Contribution

The authors propose a novel data-dependent compression approach for high-dimensional alignment, enabling existing methods to operate more efficiently with minimal quality loss.

Findings

Significant reduction in runtime with compressed data

Alignment quality comparable to original data

Framework applicable to various existing alignment methods

Abstract

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.