QUADRO: A SUPERVISED DIMENSION REDUCTION METHOD VIA RAYLEIGH QUOTIENT OPTIMIZATION

Annals of Statistics
Jianqing FanLucy Xia

Abstract

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 efficie...Continue Reading

References

Sep 17, 2005·Bioinformatics·Wenxuan ZhongYu Zhu
May 8, 2007·Bioinformatics·Zhi Wei, Hongzhe Li
Jul 27, 2007·Biometrics·Lexin Li, Xiangrong Yin
Jan 27, 2009·Annals of Statistics·Jianqing Fan, Yingying Fan
Feb 11, 2012·Journal of the Royal Statistical Society. Series B, Statistical Methodology·Daniela M Witten, Robert Tibshirani
Oct 18, 2012·Journal of the Royal Statistical Society. Series B, Statistical Methodology·Jianqing FanXin Tong
Nov 11, 2014·Journal of Computational and Graphical Statistics : a Joint Publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America·Tuo ZhaoHan Liu
Jan 20, 2015·Annals of Statistics·Jianqing FanHui Zou

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Citations

Jun 1, 2018·Nature Communications·Abubakar AbidJames Zou
Sep 15, 2018·Annals of Statistics·Jianqing FanWeichen Wang

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