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：本文提出了一种非线性降维技术——Wasserstein等距映射（Wassmap），旨在解决现有全局非线性降维算法在图像应用中的若干缺陷。Wassmap通过概率测度在Wasserstein空间中表示图像，进而利用关联测度间的成对Wasserstein距离生成低维近似等距嵌入。我们证明该算法能够精确恢复某些图像流形的参数，包括由固定生成测度的平移或伸缩变换生成的流形。此外，通过建立从函数数据到离散数据的理论桥梁，我们证明该算法的离散版本能够从离散测度生成的流形中提取参数。在多种图像数据流形上的实验表明，相较于其他全局与局部降维技术，Wassmap能生成更优的嵌入结果。