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 $\tilde{ \Omega}(n)$.
Categories: math.OC cs.DS