Linear Regression with an Unknown Permutation: Statistical and Computational Limits

TL;DR

This paper investigates the limits of recovering an unknown permutation in a noisy linear model, establishing conditions for exact and approximate recovery, and explores computational complexity with algorithms for specific cases.

Contribution

It provides sharp theoretical conditions for permutation recovery and demonstrates the NP-hardness of maximum likelihood estimation, along with a polynomial-time algorithm for the one-dimensional case.

Findings

Sharp conditions for exact permutation recovery.

NP-hardness of maximum likelihood estimation.

Polynomial-time algorithm for the case when dimension d=1.

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 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 $\Pi^*$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $\Pi^*$ is NP-hard to compute, while also providing a polynomial time algorithm when $d =1$.