On the Ranking Recovery from Noisy Observations up to a Distortion

This paper considers the problem of recovering the ranking of a data vector from noisy observations, up to a distortion. Specifically, the noisy observations consist of the original data vector corrupted by isotropic additive Gaussian noise, and the distortion is measured in terms of a distance function between the estimated ranking and the true ranking of the original data vector. First, it is shown that an optimal (in terms of error probability) decision rule for the estimation task simply outputs the ranking of the noisy observation. Then, the error probability incurred by such a decision rule is characterized in the low-noise regime, and shown to grow sublinearly with the noise standard deviation. This result highlights that the proposed approximate version of the ranking recovery problem is significantly less noise-dominated than the exact recovery considered in [Jeong, ISIT 2021].