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A tacit assumption in linear regression is that (response, predictor)-pairscorrespond to identical observational units. A series of recent works havestudied scenarios in which this assumption is violated under terms such as``Unlabeled Sensing and ``Regression with Unknown Permutation''. In this paper,we study the setup of multiple response variables and a notion of mismatchesthat generalizes permutations in order to allow for missing matches as well asfor one-to-many matches. A two-stage method is proposed under the assumptionthat most pairs are correctly matched. In the first stage, the regressionparameter is estimated by handling mismatches as contaminations, andsubsequently the generalized permutation is estimated by a basic variant ofmatching. The approach is both computationally convenient and equipped withfavorable statistical guarantees. Specifically, it is shown that the conditionsfor permutation recovery become considerably less stringent as the number ofresponses $m$ per observation increase. Particularly, for $m = \Omega(\log n)$,the required signal-to-noise ratio no longer depends on the sample size $n$.Numerical results on synthetic and real data are presented to support the mainfindings of our analysis.

A Two-Stage Approach to Multivariate Linear Regression with Sparsely Mismatched Data