Towards Optimal Transport with Global Invariances
This addresses a key limitation in machine learning for tasks involving correspondences between sets with learned representations, though it appears incremental as it extends existing optimal transport methods to handle invariances.
The paper tackles the problem of applying optimal transport when object representations have latent global transformations, such as rotations or reflections, by proposing a joint optimization framework over transport couplings and transformations, and shows promising results on tasks like unsupervised word translation.
Many problems in machine learning involve calculating correspondences between sets of objects, such as point clouds or images. Discrete optimal transport provides a natural and successful approach to such tasks whenever the two sets of objects can be represented in the same space, or at least distances between them can be directly evaluated. Unfortunately neither requirement is likely to hold when object representations are learned from data. Indeed, automatically derived representations such as word embeddings are typically fixed only up to some global transformations, for example, reflection or rotation. As a result, pairwise distances across two such instances are ill-defined without specifying their relative transformation. In this work, we propose a general framework for optimal transport in the presence of latent global transformations. We cast the problem as a joint optimization over transport couplings and transformations chosen from a flexible class of invariances, propose algorithms to solve it, and show promising results in various tasks, including a popular unsupervised word translation benchmark.