A new Linear Time Bi-level $\ell_{1,\infty}$ projection ; Application to the sparsification of auto-encoders neural networks

TL;DR

This paper introduces a novel bi-level projection method for the $\, ext{l}_{1, ext{infinity}}$ norm that significantly reduces computational complexity and enhances sparsity in neural network auto-encoders without sacrificing accuracy.

Contribution

The paper presents a new bi-level projection algorithm for the $\, ext{l}_{1, ext{infinity}}$ norm with linear time complexity and provides a mathematical identity validated through experiments.

Findings

Bi-level $\, ext{l}_{1, ext{infinity}}$ projection is 2.5 times faster than existing algorithms.

The new method achieves better sparsity while maintaining classification accuracy.

Mathematical proof and experimental validation support the new $\, ext{l}_{1, ext{infinity}}$ identity.

Abstract

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.