Linear regression without correspondence
Machine Learning
2017-11-09 v2 Statistics Theory
Machine Learning
Statistics Theory
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.
Cite
@article{arxiv.1705.07048,
title = {Linear regression without correspondence},
author = {Daniel Hsu and Kevin Shi and Xiaorui Sun},
journal= {arXiv preprint arXiv:1705.07048},
year = {2017}
}