论文标题
用于光谱限制的矩阵优化及其在公平性的应用的牛顿的立方体登记牛顿
Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness
论文作者
论文摘要
矩阵函数可用于重写光滑光谱约束的矩阵优化问题，因为在一组对称矩阵上平滑的不受约束的问题，然后通过立方注册的牛顿方法求解。事实证明，矩阵函数的二阶链条规则身份可以计算高阶导数以实现立方体规范化的牛顿，并为矩阵矢量空间的立方注册牛顿提供了新的收敛分析。我们通过对合成数据集和实际数据集进行数值实验来证明我们的方法的适用性。在我们的实验中，我们制定了一个新的模型，以估算泰勒的M-估计器（TME）模型的精神估算公平和稳健的协方差矩阵并证明其优势。
Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A second-order chain rule identity for matrix functions is proven to compute the higher-order derivatives to implement cubic-regularized Newton, and a new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces. We demonstrate the applicability of our approach by conducting numerical experiments on both synthetic and real datasets. In our experiments, we formulate a new model for estimating fair and robust covariance matrices in the spirit of the Tyler's M-estimator (TME) model and demonstrate its advantage.
