Robust matrix factor analysis method with adaptive parameter adjustment using Cauchy weighting

In recent years, high-dimensional matrix factor models have been widely applied in various fields. However, there are few methods that effectively handle heavy-tailed data. To address this problem, we introduced a smooth Cauchy loss function and established an optimization objective through norm minimization, deriving a Cauchy version of the weighted iterative estimation method. Unlike the Huber loss weighted estimation method, the weight calculation in this method is a smooth function rather than a piecewise function. It also considers the need to update parameters in the Cauchy loss function with each iteration during estimation. Ultimately, we propose a weighted estimation method with adaptive parameter adjustment. Subsequently, this paper analyzes the theoretical properties of the method, proving that it has a fast convergence rate. Through data simulation, our method demonstrates significant advantages. Thus, it can serve as a better alternative to other existing estimation methods. Finally, we analyzed a dataset of regional population movements between cities, demonstrating that our proposed method offers estimations with excellent interpretability compared to other methods.