  We consider the linear regression problem, where the number $p$ of covariates
is possibly larger than the number $n$ of observations $(x_{i},y_{i})_{i\leq i
\leq n}$, under sparsity assumptions. On the one hand, several methods have
been successfully proposed to perform this task, for example the LASSO or the
Dantzig Selector. On the other hand, consider new values $(x_{i})_{n+1\leq i
\leq m}$. If one wants to estimate the corresponding $y_{i}$'s, one should
think of a specific estimator devoted to this task, referred by Vapnik as a
"transductive" estimator. This estimator may differ from an estimator designed
to the more general task "estimate on the whole domain". In this work, we
propose a generalized version both of the LASSO and the Dantzig Selector, based
on the geometrical remarks about the LASSO in pr\'evious works. The "usual"
LASSO and Dantzig Selector, as well as new estimators interpreted as
transductive versions of the LASSO, appear as special cases. These estimators
are interesting at least from a theoretical point of view: we can give
theoretical guarantees for these estimators under hypotheses that are relaxed
versions of the hypotheses required in the papers about the "usual" LASSO.
These estimators can also be efficiently computed, with results comparable to
the ones of the LASSO.
