In this paper, we carry out a unified study for $L_1$ over $L_2$ sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signals. First, we provide a unified theoretical analysis on the existence of the global solutions of the constrained and the unconstrained $L_{1}/L_{2}$ models. Second, we analyze the sparse property of any local minimizer of these $L_{1}/L_{2}$ models which serves as a certificate to rule out the nonlocal-minimizer stationary solutions. Third, we derive an analytical solution for the proximal operator of the $L_{1} / L_{2}$ with nonnegative constraint. Equipped with this, we apply the alternating direction method of multipliers to the unconstrained model with nonnegative constraint in a particular splitting way, referred to as ADMM$_p^+$. We establish its global convergence to a d-stationary solution (sharpest stationary) without the Kurdyka-\L ojasiewicz assumption. Extensive numerical simulations confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. In particular, ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy than the commonly used scaled gradient projection method for the wavelength misalignment problem.
翻译:在本文中,我们为超过1美元L1}/L2}美元的地方最小化模型的稀有性质进行了统一研究,这些模型被广泛用于恢复稀少的非消极/任意信号的统一字典制度。首先,我们对限制和不受限制的$L1}/L2}美元模型的全球解决方案的存在情况进行了统一的理论分析。第二,我们分析了这些$1}/L2}美元模型的本地最小化模型的稀薄性质,作为排除非当地最低货币固定解决方案的证书。第三,我们为美元/L1}/L2}的准字典操作者提出了一个分析解决方案,而没有消极限制。为此,我们用一种交替式的乘数方法对不受限制的模型进行了交替式指导,特别称之为ADMM$_p$/美元。我们建立了全球趋同一种固定解决方案(固定固定状态)的结合,而没有Kurdyka-L$/L2}/L2} 美元无偏差限制的准操作者,同时用ADM_M_ 通常的货币回收方法来确认最高货币的精确度假设。