Linear regression without correspondence

TL;DR

This paper explores algorithms for linear regression when the correspondence between data points and responses is unknown, providing efficient solutions in certain settings and establishing fundamental limits on recovery.

Contribution

It introduces a polynomial-time approximation scheme for the least squares problem without correspondence and an exact recovery algorithm in a noise-free, average-case scenario.

Findings

Polynomial-time approximation scheme for least squares without correspondence

Efficient lattice-based algorithm for exact recovery in noise-free case

Lower bounds on signal-to-noise ratio for approximate recovery

Abstract

This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown. First, a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension. Next, in an average-case and noise-free setting where the responses exactly correspond to a linear function of i.i.d. draws from a standard multivariate normal distribution, an efficient algorithm based on lattice basis reduction is shown to exactly recover the unknown linear function in arbitrary dimension. Finally, lower bounds on the signal-to-noise ratio are established for approximate recovery of the unknown linear function by any estimator.