  We derive expressions for the finite-sample distribution of the Lasso
estimator in the context of a linear regression model in low as well as in high
dimensions by exploiting the structure of the optimization problem defining the
estimator. In low dimensions, we assume full rank of the regressor matrix and
present expressions for the cumulative distribution function as well as the
densities of the absolutely continuous parts of the estimator. Our results are
presented for the case of normally distributed errors, but do not hinge on this
assumption and can easily be generalized. Additionally, we establish an
explicit formula for the correspondence between the Lasso and the least-squares
estimator. We derive analogous results for the distribution in less explicit
form in high dimensions where we make no assumptions on the regressor matrix at
all. In this setting, we also investigate the model selection properties of the
Lasso and show that possibly only a subset of models might be selected by the
estimator, completely independently of the observed response vector. Finally,
we present a condition for uniqueness of the estimator that is necessary as
well as sufficient.
