In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a nonlinear dimensionality reduction technique that provides solutions to some drawbacks in existing global nonlinear dimensionality reduction algorithms in imaging applications. Wassmap represents images via probability measures in Wasserstein space, then uses pairwise Wasserstein distances between the associated measures to produce a low-dimensional, approximately isometric embedding. We show that the algorithm is able to exactly recover parameters of some image manifolds including those generated by translations or dilations of a fixed generating measure. Additionally, we show that a discrete version of the algorithm retrieves parameters from manifolds generated from discrete measures by providing a theoretical bridge to transfer recovery results from functional data to discrete data. Testing of the proposed algorithms on various image data manifolds show that Wassmap yields good embeddings compared with other global and local techniques.

翻译：在本文中,我们提议瓦西斯坦测量图(Wassmap),这是一种非线性维度减少技术,它为现有全球非线性降低成像应用算法的某些缺点提供了解决办法。Wassmap代表了瓦西斯坦空间中通过概率测量得出的图像,然后使用对称瓦西斯坦相距相关测量法来产生一个低维度的大致等度嵌入。我们表明,算法能够准确恢复一些图像元的参数,包括固定生成测量的翻译或变相所产生的参数。此外,我们表明,一个离散的算法版本通过提供理论桥梁将功能数据恢复结果转换为离散数据,从离散测量生成的元中提取参数。对各种图像数据元的拟议的算法进行的测试表明,相对于其他全球和本地技术,瓦西马相生成了良好的嵌入。