Partial Wasserstein Covering

We consider a general task called partial Wasserstein covering with the goal of emulating a large dataset (e.g., application dataset) using a small dataset (e.g., development dataset) in terms of the empirical distribution by selecting a small subset from a candidate dataset and adding it to the small dataset. We model this task as a discrete optimization problem with partial Wasserstein divergence as an objective function. Although this problem is NP-hard, we prove that it has the submodular property, allowing us to use a greedy algorithm with a 0.63 approximation. However, the greedy algorithm is still inefficient because it requires linear programming for each objective function evaluation. To overcome this difficulty, we propose quasi-greedy algorithms for acceleration, which consist of a series of techniques such as sensitivity analysis based on strong duality and the so-called $C$-transform in the optimal transport field. Experimentally, we demonstrate that we can efficiently make two datasets similar in terms of partial Wasserstein divergence, including driving scene datasets.