Matching Map Recovery with an Unknown Number of Outliers

• URL: http://arxiv.org/abs/2210.13354v1
• Date: Mon, 24 Oct 2022 15:59:10 GMT
• Title: Matching Map Recovery with an Unknown Number of Outliers
• Authors: Arshak Minasyan, Tigran Galstyan, Sona Hunanyan, Arnak Dalalyan
• Abstract summary: We consider the problem of finding the matching map between two sets of $d$-dimensional noisy feature-vectors. We show that, in the high-dimensional setting, if the signal-to-noise ratio is larger than $5(dlog(4nm/alpha))1/4$, the true matching map can be recovered with probability $1-alpha$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of finding the matching map between two sets of $d$-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 $n$ and $m$ 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 $k^*\le \min(n,m)$. We show that, in the high-dimensional setting, if the signal-to-noise ratio is larger than $5(d\log(4nm/\alpha))^{1/4}$, then the true matching map can be recovered with probability $1-\alpha$. Interestingly, this threshold does not depend on $k^*$ and is the same as the one obtained in prior work in the case of $k = \min(n,m)$. The procedure for which the aforementioned property is proved is obtained by a data-driven selection among candidate mappings $\{\hat\pi_k:k\in[\min(n,m)]\}$. Each $\hat\pi_k$ minimizes the sum of squares of distances between two sets of size $k$. 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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