We propose a novel differentiable reformulation of the linearly-constrained \(\ell _1\) minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based methods for the solution of \(\ell _1\) minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The results can be seen as bounds on the complexity of iteratively reweighted least squares (IRLS) type methods of basis pursuit.

我们提出了线性约束\（\ ell _1 \）最小化问题的一种新的可微重构形式，也称为基础追求问题。重新制定受到网络理论的拉普拉斯范式的启发，并导致了一系列新的基于梯度的方法来解决\（\ ell _1 \）最小化问题。我们基于乘性权重更新方案以及加速梯度方案的迭代复杂度，分析了自然求解方法的迭代复杂度。结果可以看成是基于迭代的加权最小二乘（IRLS）类型的基本追踪方法的复杂性的界限。