Lagrange对偶框架下的重加权$\ell_1$算法：赋予权重可解释性
Re-Weighted $\ell_1$ Algorithms within the Lagrange Duality Framework: Bringing Interpretability to Weights
摘要：
本文研究信号处理中的一个重要问题，即求线性系$Φx=b$的稀疏解。这个问题在许多领域都有应用，但一般来说是NP困难的。通常，基于最小化（加权）$\ell_{1}$范数来考虑备选凸问题。为了使这种替代方案有用，应该选择权重以获得原始NP难问题的解决方案。一个众所周知的算法是重新加权$\ell_{1}$，由Candès，Wakin和Boyd提出。在这篇文章中，我们介绍了一种新的方法来更新的权重的重新加权$\ell_{1}$算法，根据确定这些权重的拉格朗日乘子。然后，这被翻译成一个算法的性能与通常的方法，但允许解释的权重为拉格朗日乘数。该方法也可以用于噪声线性系，在这种情况下获得重加权LASSO算法，根据实验结果具有有希望的性能。
英文摘要：
We consider an important problem in signal processing, which consists in finding the sparsest solution of a linear system $Φx=b$. This problem has applications in several areas, but is NP-hard in general. Usually an alternative convex problem is considered, based on minimizing the (weighted) $\ell_{1}$ norm. For this alternative to be useful, weights should be chosen as to obtain a solution of the original NP-hard problem. A well known algorithm for this is the Re-Weighted $\ell_{1}$, proposed by Candès, Wakin and Boyd. In this article we introduce a new methodology for updating the weights of a Re-Weighted $\ell_{1}$ algorithm, based on identifying these weights as Lagrange multipliers. This is then translated into an algorithm with performance comparable to the usual methodology, but allowing an interpretation of the weights as Lagrange multipliers. The methodology may also be used for a noisy linear system, obtaining in this case a Re-Weighted LASSO algorithm, with a promising performance according to the experimental results.
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