QUADRO: A supervised dimension reduction method via Rayleigh quotient optimization
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QUADRO：基于瑞利商优化的监督降维
- 研究问题
- 如何对高维数据进行稀疏二次监督降维，并处理瑞利商优化所涉及的大量四阶交叉矩及重尾数据带来的估计误差？
- 方法与思路
- 作者在椭圆分布模型下简化二次统计量的方差表达，以稳健均值和协方差估计构造凸优化问题，并用线性化增广拉格朗日法求解。
- 材料陈述的贡献
- 摘要提出 QUADRO，区分非线性情形下瑞利商优化与分类，并报告高斯及一般椭圆模型中的瑞利商收敛速率和数值验证。
- 阅读关注点
- 应核对椭圆分布、稀疏性和四阶矩条件如何发挥作用，优化目标与分类损失的差别，以及算法收敛和统计误差保证的范围。
- 阅读边界
- 所据为 arXiv 预印本，同行评审与期刊接收未核实；摘要未给出具体速率或实验数值，不能推断其适用于任意重尾分布。
原文摘要
We propose a novel Rayleigh quotient based sparse quadratic dimension reduction method - named QUADRO (Quadratic Dimension Reduction via Rayleigh Optimization) - for analyzing high- dimensional data. Unlike in the linear setting where Rayleigh quotient optimization coincides with classification, these two problems are very different under nonlinear settings. In this paper, we clarify this difference and show that Rayleigh quotient optimization may be of independent scientific interests. One major challenge of Rayleigh quotient optimization is that the variance of quadratic statistics involves all fourth cross-moments of predictors, which are infeasible to compute for high-dimensional applications and may accumulate too many stochastic errors. This issue is resolved by considering a family of elliptical models. Moreover, for heavy-tail distributions, robust estimates of mean vectors and covariance matrices are employed to guarantee uniform convergence in estimating nonpolynomially many parameters, even though only the fourth moments are assumed. Methodologically, QUADRO is based on elliptical models which allow us to formulate the Rayleigh quotient maximization as a convex optimization problem. Computationally, we propose an efficient linearized augmented Lagrangian method to solve the constrained optimization problem. Theoretically, we provide explicit rates of convergence in terms of Rayleigh quotient under both Gaussian and general elliptical models. Thorough numerical results on both synthetic and real datasets are also provided to back up our theoretical results.