  The $\ell_{1,\infty}$ norm is an efficient-structured projection, but the
complexity of the best algorithm is, unfortunately, $\mathcal{O}\big(n m \log(n
m)\big)$ for a matrix $n\times m$.\\ In this paper, we propose a new bi-level
projection method, for which we show that the time complexity for the
$\ell_{1,\infty}$ norm is only $\mathcal{O}\big(n m \big)$ for a matrix
$n\times m$. Moreover, we provide a new $\ell_{1,\infty}$ identity with
mathematical proof and experimental validation. Experiments show that our
bi-level $\ell_{1,\infty}$ projection is $2.5$ times faster than the actual
fastest algorithm and provides the best sparsity while keeping the same
accuracy in classification applications.
