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

Understand

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 $\mathbb{R}^2$ 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 $k \ge 0$, called $k$-RPW that is based on computing the partial $2$-Wasserstein distance.