Linear Regression With Shuffled Data: Statistical and Computational Limits of Permutation Recovery

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 x 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 matrix A are drawn independently from a standard Gaussian distribution and establish sharp conditions on the signal-to-noise ratio, 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 for general d, while also providing a polynomial time algorithm when d = 1.