QUADRO: A SUPERVISED DIMENSION REDUCTION METHOD VIA RAYLEIGH QUOTIENT OPTIMIZATION.
QUADRO: A SUPERVISED DIMENSION REDUCTION METHOD VIA RAYLEIGH QUOTIENT OPTIMIZATION.
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DOI：
10.1214/14-aos1307
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发表时间：
2015
影响因子：
4.5
通讯作者：
Xia L
中科院分区：
文献类型：
--
作者：
Fan J;Ke ZT;Liu H;Xia L
我们提出了一种基于瑞利商的新型稀疏二次尺寸减少方法 - 命名Quadro（通过瑞利优化的二次尺寸降低），用于分析的高维数据。在本文的非线性设置下，我们澄清了这一差异优化可能是独立的科学利益通过考虑椭圆模型的家庭来解决。即使仅假定第四次矩，Quadro是基于椭圆形模型，从而确保估算非多种参数的统一收敛有效的线性增强Lagrangian方法解决了约束优化问题。在高斯和一般椭圆模型下，瑞利商的术语也提供了合成和真实数据集的彻底数值结果。
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 non-polynomially 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.