This paper applies an idea of adaptive momentum for the nonlinear conjugate gradient to accelerate optimization problems in sparse recovery. Specifically, we consider two types of minimization problems: a (single) differentiable function and the sum of a non-smooth function and a differentiable function. In the first case, we adopt a fixed step size to avoid the traditional line search and establish the convergence analysis of the proposed algorithm for a quadratic problem. This acceleration is further incorporated with an operator splitting technique to deal with the non-smooth function in the second case. We use the convex $\ell_1$ and the nonconvex $\ell_1-\ell_2$ functionals as two case studies to demonstrate the efficiency of the proposed approaches over traditional methods.

翻译：本文提出将自适应动量思想应用于非线性共轭梯度，以加速稀疏恢复中的优化问题。具体而言，我们考虑两类最小化问题：可微函数（单目标）以及非光滑函数与可微函数之和。对于第一类问题，我们采用固定步长以避免传统线搜索，并建立所提算法在二次问题上的收敛性分析。该加速方法进一步与算子分裂技术相结合，以处理第二类问题中的非光滑函数。我们以凸$\ell_1$泛函和非凸$\ell_1-\ell_2$泛函作为两个案例研究，证明所提方法相较于传统方法的有效性。