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

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> , the probability of error is shown to behave proportionally to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sigma$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sigma$</tex> is the noise standard deviation. Moreover, the slope is computed exactly for several distributions and it is shown to behave quadratically in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> . In the high noise regime, for every dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> , the probability of correctness is shown to behave as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1/\sigma$</tex> , and the exact expression for the rate of convergence is also provided.