Joint Correlation Detection and Alignment of Gaussian Databases
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In this work, we propose an efficient two-stage algorithm solving a joint problem of correlation detection and permutation recovery between two Gaussian databases. Correlation detection is an hypothesis testing problem; under the null hypothesis, the databases are independent, and under the alternate hypothesis, they are correlated, under an unknown row permutation. We develop relatively tight bounds on the type-I and type-II error probabilities, and show that the analyzed detector performs better than a recently proposed detector, at least for some specific parameter choices. Since the proposed detector relies on a statistic, which is a sum of dependent indicator random variables, then in order to bound the type-I probability of error, we develop a novel graph-theoretic technique for bounding the k-th order moments of such statistics. When the databases are accepted as correlated, the algorithm also outputs an estimation for the underlying row permutation. By comparing to known converse results for this problem, we prove that the alignment error probability converges to zero under the asymptotically lowest possible correlation coefficient.
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