Linear regression with an unknown permutation: Statistical and computational limits

Consider 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 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.