Retrieving Data Permutations from Noisy Observations: High and Low Noise Asymptotics

This paper considers the problem of recovering the permutation of an n-dimensional random vector X observed in Gaussian noise. First, a general expression for the probability of error is derived when a linear decoder (i.e., linear estimator followed by a sorting operation) is used. The derived expression holds with minimal assumptions on the distribution of X and when the noise has memory. Second, for the case of isotropic noise (i.e., noise with a diagonal scalar covariance matrix), the rates of convergence of the probability of error are characterized in the high and low noise regimes. In the low noise regime, for every dimension
$n$
, the probability of error is shown to behave proportionally to
$\sigma$
, where
$\sigma$
is the noise standard deviation. Moreover, the slope is computed exactly for several distributions and it is shown to behave quadratically in
$n$
. In the high noise regime, for every dimension
$n$
, the probability of correctness is shown to behave as
$1/\sigma$
, and the exact expression for the rate of convergence is also provided.