Linear regression without correspondence
This addresses a fundamental problem in statistics and machine learning where data pairing is missing, offering theoretical guarantees for recovery.
This paper tackles linear regression when the correspondence between covariates and responses is unknown, providing an approximation scheme for constant dimensions, an efficient algorithm for exact recovery in noise-free settings, and lower bounds on signal-to-noise ratio for approximate recovery.
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.