  A tacit assumption in linear regression is that (response, predictor)-pairs
correspond to identical observational units. A series of recent works have
studied 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 mismatches
that generalizes permutations in order to allow for missing matches as well as
for one-to-many matches. A two-stage method is proposed under the assumption
that most pairs are correctly matched. In the first stage, the regression
parameter is estimated by handling mismatches as contaminations, and
subsequently the generalized permutation is estimated by a basic variant of
matching. The approach is both computationally convenient and equipped with
favorable statistical guarantees. Specifically, it is shown that the conditions
for permutation recovery become considerably less stringent as the number of
responses $m$ 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 main
findings of our analysis.
