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 expre...

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Bibliographic Details
Published in2021 IEEE International Symposium on Information Theory (ISIT) pp. 1100 - 1105
Main Authors Jeong, Minoh, Dytso, Alex, Cardone, Martina
Format Conference Proceeding
LanguageEnglish
Published IEEE 12.07.2021
Subjects
Convergence
Covariance matrices
Decoding
Gaussian noise
Genetic expression
Noise measurement
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Summary: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.
DOI:10.1109/ISIT45174.2021.9518137