Near-Linear Time Projection onto the $\ell_{1,\infty}$ Ball; Application
  to Sparse Autoencoders
Guillaume Perez Laurent Condat Michel Barlaud
Looking for sparsity is nowadays crucial to speed up the training of
large-scale neural networks. Projections onto the $\ell_{1,2}$ and
$\ell_{1,\infty}$ are among the most efficient techniques to sparsify and
reduce the overall cost of neural networks. In this paper, we introduce a new
projection algorithm for the $\ell_{1,\infty}$ norm ball. The worst-case time
complexity of this algorithm is $\mathcal{O}\big(nm+J\log(nm)\big)$ for a
matrix in $\mathbb{R}^{n\times m}$. $J$ is a term that tends to 0 when the
sparsity is high, and to $nm$ when the sparsity is low. Its implementation is
easy and it is guaranteed to converge to the exact solution in a finite time.
Moreover, we propose to incorporate the $\ell_{1,\infty}$ ball projection while
training an autoencoder to enforce feature selection and sparsity of the
weights. Sparsification appears in the encoder to primarily do feature
selection due to our application in biology, where only a very small part
($<2\%$) of the data is relevant. We show that both in the biological case and
in the general case of sparsity that our method is the fastest.