  Consider the regression problem where the response $Y\in\mathbb{R}$ and the
covariate $X\in\mathbb{R}^d$ for $d\geq 1$ are \textit{unmatched}. Under this
scenario, we do not have access to pairs of observations from the distribution
of $(X, Y)$, but instead, we have separate datasets $\{Y_i\}_{i=1}^n$ and
$\{X_j\}_{j=1}^m$, possibly collected from different sources. We study this
problem assuming that the regression function is linear and the noise
distribution is known or can be estimated. We introduce an estimator of the
regression vector based on deconvolution and demonstrate its consistency and
asymptotic normality under an identifiability assumption. In the general case,
we show that our estimator (DLSE: Deconvolution Least Squared Estimator) is
consistent in terms of an extended $\ell_2$ norm. Using this observation, we
devise a method for semi-supervised learning, i.e., when we have access to a
small sample of matched pairs $(X_k, Y_k)$. Several applications with synthetic
and real datasets are considered to illustrate the theory.
