A Laplacian approach to $$\ell _1$$-norm minimization

We propose a novel differentiable reformulation of the linearly-constrained $\ell_1$ minimization problem, also known as basis pursuit problem. The is inspired by Laplacian paradigm network theory and leads to new family gradient-based methods for solution problems. analyze iteration complexity natural approach reformulation, based on multiplicative weights update scheme, well an accelerated gradient scheme. results can be seen bounds iteratively reweighted least squares (IRLS) type pursuit.