  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
