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Matching Map Recovery with an Unknown Number of Outliers

Abstract

We consider the problem of finding the matching map between two sets of dd-dimensional noisy feature-vectors. The distinctive feature of our setting is that we do not assume that all the vectors of the first set have their corresponding vector in the second set. If nn and mm are the sizes of these two sets, we assume that the matching map that should be recovered is defined on a subset of unknown cardinality kmin(n,m)k^*\le \min(n,m). We show that, in the high-dimensional setting, if the signal-to-noise ratio is larger than 5(dlog(4nm/α))1/45(d\log(4nm/\alpha))^{1/4}, then the true matching map can be recovered with probability 1α1-\alpha. Interestingly, this threshold does not depend on kk^* and is the same as the one obtained in prior work in the case of k=min(n,m)k = \min(n,m). The procedure for which the aforementioned property is proved is obtained by a data-driven selection among candidate mappings {π^k:k[min(n,m)]}\{\hat\pi_k:k\in[\min(n,m)]\}. Each π^k\hat\pi_k minimizes the sum of squares of distances between two sets of size kk. The resulting optimization problem can be formulated as a minimum-cost flow problem, and thus solved efficiently. Finally, we report the 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 algorithms studied in this work.

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