Quotient regularization models (QRMs) are a class of powerful regularization techniques that have gained considerable attention in recent years, due to their ability to handle complex and highly nonlinear data sets. However, the nonconvex nature of QRM poses a significant challenge in finding its optimal solution. We are interested in scenarios where both the numerator and the denominator of QRM are absolutely one-homogeneous functions, which is widely applicable in the fields of signal processing and image processing. In this paper, we utilize a gradient flow to minimize such QRM in combination with a quadratic data fidelity term. Our scheme involves solving a convex problem iteratively.The convergence analysis is conducted on a modified scheme in a continuous formulation, showing the convergence to a stationary point. Numerical experiments demonstrate the effectiveness of the proposed algorithm in terms of accuracy, outperforming the state-of-the-art QRM solvers.

翻译：商正则化模型（QRM）是一类强大的正则化技术，因其能处理复杂且高度非线性数据集而在近年来备受关注。然而，QRM的非凸性特征给寻找其最优解带来了重大挑战。本文关注QRM中分子与分母均为绝对一次齐次函数的情形，该情形在信号处理和图像处理领域具有广泛适用性。我们采用梯度流方法，结合二次数据保真项来最小化此类QRM，其求解方案需通过迭代方式解决凸优化问题。基于连续形式的修正方案进行的收敛性分析表明，该方法可收敛至驻点。数值实验验证了所提算法在精度方面的有效性，其性能优于当前最优的QRM求解器。