  In many machine learning applications, it is necessary to meaningfully
aggregate, through alignment, different but related datasets. Optimal transport
(OT)-based approaches pose alignment as a divergence minimization problem: the
aim is to transform a source dataset to match a target dataset using the
Wasserstein distance as a divergence measure. We introduce a hierarchical
formulation of OT which leverages clustered structure in data to improve
alignment in noisy, ambiguous, or multimodal settings. To solve this
numerically, we propose a distributed ADMM algorithm that also exploits the
Sinkhorn distance, thus it has an efficient computational complexity that
scales quadratically with the size of the largest cluster. When the
transformation between two datasets is unitary, we provide performance
guarantees that describe when and how well aligned cluster correspondences can
be recovered with our formulation, as well as provide worst-case dataset
geometry for such a strategy. We apply this method to synthetic datasets that
model data as mixtures of low-rank Gaussians and study the impact that
different geometric properties of the data have on alignment. Next, we applied
our approach to a neural decoding application where the goal is to predict
movement directions and instantaneous velocities from populations of neurons in
the macaque primary motor cortex. Our results demonstrate that when clustered
structure exists in datasets, and is consistent across trials or time points, a
hierarchical alignment strategy that leverages such structure can provide
significant improvements in cross-domain alignment.
