  We consider the problem of linear regression where the $\ell_2^n$ norm loss
(i.e., the usual least squares loss) is replaced by the $\ell_p^n$ norm. We
show how to solve such problems up to machine precision in $O^*(n^{|1/2 -
1/p|})$ (dense) matrix-vector products and $O^*(1)$ matrix inversions, or
alternatively in $O^*(n^{|1/2 - 1/p|})$ calls to a (sparse) linear system
solver. This improves the state of the art for any $p\not\in \{1,2,+\infty\}$.
Furthermore we also propose a randomized algorithm solving such problems in
{\em input sparsity time}, i.e., $O^*(Z + \mathrm{poly}(d))$ where $Z$ is the
size of the input and $d$ is the number of variables. Such a result was only
known for $p=2$. Finally we prove that these results lie outside the scope of
the Nesterov-Nemirovski's theory of interior point methods by showing that any
symmetric self-concordant barrier on the $\ell_p^n$ unit ball has
self-concordance parameter $\tilde{\Omega}(n)$.
