Near-Linear Time Projection onto the $\ell_{1,\infty}$ Ball; Application to Sparse Autoencoders

TL;DR

This paper introduces a fast, exact projection algorithm onto the $\, ext{l}_{1, ext{infinity}}$ norm ball with applications to training sparse autoencoders, significantly improving efficiency in enforcing sparsity in neural networks.

Contribution

A novel projection algorithm for the $\, ext{l}_{1, ext{infinity}}$ norm ball with near-linear time complexity and practical implementation for neural network sparsification.

Findings

Algorithm is the fastest for sparsity enforcement.

Exact projection guarantees convergence in finite time.

Application to biological data demonstrates effectiveness.

Abstract

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.