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. 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 $\mathbb{R}^2$
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 $p
\in [1,\infty]$. 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.

给定一篇研究论文，提取五个关键词来准确表示其主要主题和研究领域，然后用一句简洁的中文句子对论文进行总结。