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

Abstract

Consider a noisy linear observation model with an unknown permutation, based on observing $y = \Pi ^{*} A x^{*} + w$ , where $x^{*} \in {\mathbb {R}} ^{d}$ is an unknown vector, $\Pi ^{*}$ is an unknown $n \times n$ permutation matrix, and $w \in {\mathbb {R}} ^{n}$ 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 $\Pi ^{*}$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $\Pi ^{*}$ is NP-hard to compute for general $d$ , while also providing a polynomial time algorithm when $d =1$ .