Steps beyond this constraint include using small amounts of parallel data (Zhang et al., 2016) or an unsupervised technique as the initial step to generate pseudo-parallel data (Conneau et al., 2018) before solving for P. Optimal transport formalizes the problem of finding a minimum cost mapping between two point sets, viewed as discrete distributions.
we assume two empirical distributions over embeddings, e.g., where p and q are vectors of probability weights associated with each point set.