Near-Linear Time Projection onto the $\ell_{1,\infty}$ Ball; Application to Sparse Autoencoders
This work addresses the need for faster sparsification techniques in large-scale neural networks, particularly for applications like biology where data relevance is low, though it is incremental as it builds on existing projection methods.
The paper tackles the problem of efficiently projecting onto the $\ell_{1,\infty}$ norm ball to sparsify neural networks, introducing an algorithm with worst-case time complexity $\mathcal{O}(nm + J\log(nm))$ that is guaranteed to converge exactly and is applied to autoencoders for feature selection, showing it is the fastest method in both biological and general cases.
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.