Linear regression with an unknown permutation: Statistical and computational limits
A Pananjady, MJ Wainwright… - 2016 54th Annual …, 2016 - ieeexplore.ieee.org
A Pananjady, MJ Wainwright, TA Courtade
2016 54th Annual Allerton Conference on Communication, Control …, 2016•ieeexplore.ieee.orgConsider a noisy linear observation model with an unknown permutation, based on
observing y= Π* Ax*+ w, where x*∈ ℝ d is an unknown vector, Π* is an unknown n× n
permutation matrix, and w∈ ℝ n is additive Gaussian noise. We analyze the problem of
permutation recovery in a random design setting in which the entries of the matrix A are
drawn iid from a standard Gaussian distribution, and establish sharp conditions on the SNR,
sample size n, and dimension d under which Π* is exactly and approximately recoverable …
observing y= Π* Ax*+ w, where x*∈ ℝ d is an unknown vector, Π* is an unknown n× n
permutation matrix, and w∈ ℝ n is additive Gaussian noise. We analyze the problem of
permutation recovery in a random design setting in which the entries of the matrix A are
drawn iid from a standard Gaussian distribution, and establish sharp conditions on the SNR,
sample size n, and dimension d under which Π* is exactly and approximately recoverable …
Consider a noisy linear observation model with an unknown permutation, based on observing y = Π*Ax* + w, where x* ∈ ℝd is an unknown vector, Π* is an unknown n × n permutation matrix, and w ∈ ℝn is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of the matrix A are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size n, and dimension d under which Π* is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of Π* is NP-hard to compute, while also providing a polynomial time algorithm when d = 1.
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