  We consider the problem of finding the matching map between two sets of $d$
dimensional vectors from noisy observations, where the second set contains
outliers. The matching map is then an injection, which can be consistently
estimated only if the vectors of the second set are well separated. The main
result shows that, in the high-dimensional setting, a detection region of
unknown injection can be characterized by the sets of vectors for which the
inlier-inlier distance is of order at least $d^{1/4}$ and the inlier-outlier
distance is of order at least $d^{1/2}$. These rates are achieved using the
estimated matching minimizing the sum of logarithms of distances between
matched pairs of points. We also prove lower bounds establishing optimality of
these rates. Finally, we report results of numerical experiments on both
synthetic and real world data that illustrate our theoretical results and
provide further insight into the properties of the estimators studied in this
work.
