Linear Regression with an Unknown Permutation: Statistical and Computational Limits

Abstract

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

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