We introduce Joint Multidimensional Scaling, a novel approach for unsupervised manifold alignment, which maps datasets from two different domains, without any known correspondences between data instances across the datasets, to a common low-dimensional Euclidean space. Our approach integrates Multidimensional Scaling (MDS) and Wasserstein Procrustes analysis into a joint optimization problem to simultaneously generate isometric embeddings of data and learn correspondences between instances from two different datasets, while only requiring intra-dataset pairwise dissimilarities as input. This unique characteristic makes our approach applicable to datasets without access to the input features, such as solving the inexact graph matching problem. We propose an alternating optimization scheme to solve the problem that can fully benefit from the optimization techniques for MDS and Wasserstein Procrustes. We demonstrate the effectiveness of our approach in several applications, including joint visualization of two datasets, unsupervised heterogeneous domain adaptation, graph matching, and protein structure alignment. The implementation of our work is available at https://github.com/BorgwardtLab/JointMDS

翻译：我们提出联合多维缩放（Joint Multidimensional Scaling），一种新颖的无监督流形对齐方法，该方法将来自两个不同领域的数据集映射到共同的低维欧几里得空间，而无需数据实例间的任何已知对应关系。我们的方法将多维缩放（MDS）和Wasserstein Procrustes分析整合为一个联合优化问题，在仅需数据内部成对相异性作为输入的情况下，同时生成数据的等距嵌入并学习两个不同数据集实例间的对应关系。这一独特特性使我们的方法适用于无法访问输入特征的数据集，例如解决非精确图匹配问题。我们提出一种交替优化方案来求解该问题，该方案能充分利用MDS和Wasserstein Procrustes的优化技术。我们在多个应用中展示了方法的有效性，包括两个数据集的联合可视化、无监督异构域适应、图匹配以及蛋白质结构对齐。本工作的实现代码见https://github.com/BorgwardtLab/JointMDS。