We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, under a weak \textit{locally bounded non-convexity condition}, we prove that the optimality of coordinate-wise stationary point is stronger than that of the standard critical point and directional point. Under additional suitable conditions, CD methods converge Q-linearly to coordinate-wise stationary points. In the case of a concave denominator, we show that any critical point is a global minimum, and CD methods converge to the global minimum with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.

翻译：我们考虑一类结构化分数最小化问题，其目标函数的分子部分是可微凸函数与凸非光滑函数之和，分母部分为凸函数或凹函数。由于该问题非凸，求解难度较大。通过利用问题结构，我们提出两种坐标下降（CD）方法用于求解。所提方法迭代全局求解一维子问题，并保证收敛至坐标稳定点。在分母为凸函数的情形下，基于弱局部有界非凸性条件，我们证明坐标稳定点的最优性优于标准临界点与方向点。在附加适当条件下，CD方法以Q线性收敛速度收敛至坐标稳定点。在分母为凹函数的情形下，我们证明任意临界点均为全局最小值，且CD方法以次线性收敛速度收敛至全局最小值。我们展示了所提方法在若干机器学习与信号处理模型中的适用性。基于真实数据的实验表明，我们的方法在精度上显著且一致地优于现有方法。