A New Robust Partial p-Wasserstein-Based Metric for Comparing Distributions

The $2$-Wasserstein distance is sensitive to minor geometric differences between distributions, making it a very powerful dissimilarity metric. However, due to this sensitivity, a small outlier mass can also cause a significant increase in the $2$-Wasserstein distance between two similar distributions. Similarly, sampling discrepancy can cause the empirical $2$-Wasserstein distance on n samples in R² to converge to the true distance at a rate of n-1/4, which is significantly slower than the rate of n-1/2 for $1$-Wasserstein distance. We introduce a new family of distances parameterized by k ≥ 0, called k-RPW, that is based on computing the partial $2$-Wasserstein distance. We show that (1) k-RPW satisfies the metric properties, (2) k-RPW is robust to small outlier mass while retaining the sensitivity of $2$-Wasserstein distance to minor geometric differences, and (3) when k is a constant, k-RPW distance between empirical distributions on n samples in R² converges to the true distance at a rate of n-1/3, which is faster than the convergence rate of n-1/4 for the $2$-Wasserstein distance. Using the partial p-Wasserstein distance, we extend our distance to any p ∈ [1,∞]. By setting parameters k or p appropriately, we can reduce our distance to the total variation, p-Wasserstein, and the L\'evy-Prokhorov distances. Experiments show that our distance function achieves higher accuracy in comparison to the $1$-Wasserstein, $2$-Wasserstein, and TV distances for image retrieval tasks on noisy real-world data sets.