  We consider the task of aligning two sets of points in high dimension, which
has many applications in natural language processing and computer vision. As an
example, it was recently shown that it is possible to infer a bilingual
lexicon, without supervised data, by aligning word embeddings trained on
monolingual data. These recent advances are based on adversarial training to
learn the mapping between the two embeddings. In this paper, we propose to use
an alternative formulation, based on the joint estimation of an orthogonal
matrix and a permutation matrix. While this problem is not convex, we propose
to initialize our optimization algorithm by using a convex relaxation,
traditionally considered for the graph isomorphism problem. We propose a
stochastic algorithm to minimize our cost function on large scale problems.
Finally, we evaluate our method on the problem of unsupervised word
translation, by aligning word embeddings trained on monolingual data. On this
task, our method obtains state of the art results, while requiring less
computational resources than competing approaches.
