Sparse linear discriminant analysis for simultaneous testing for the significance of a gene set/pathway and gene selection

Wu, Michael C.; Zhang, Lingsong; Wang, Zhao‐Xi; Christiani, David C.; Lin, Xihong

doi:10.1093/bioinformatics/btp019

Cited by 102 publications

(99 citation statements)

References 27 publications

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“…It is also interesting to note that, in the case D = W , SCF requires about only three iterations to achieve a highly accurate solution (with residual ϱ up to 10 −9 ) for a random problem (1.1). In addition, we also observed that, except for the case n = 100, all these algorithms approach the same objective value for each test problem; in several test 4 In the MATLAB environment, the generic Riemannian trust-region package for the optimization of functions defined on Riemannian manifolds is available at http://www.math.fsu.edu/~cbaker/GenRTR/ 5 The MATLAB code sg_min of the latest version (version 2.4.3) is available at http://web.mit.edu/~ripper/www/sgmin.html 6 Because sg_minminimizes a function on the Stiefel manifold, we use −f (x) instead of f (x). 7 We point out that it does not lose any generality to assume that B is also positive definite as we can shift B to B + ξ W so that it is positive definite but with the maximizers unchanged.…”

Section: Preliminary Numerical Experimentsmentioning

confidence: 60%

“…In theory, one can assume, without loss of generality, that D is also positive definite and even diagonal because of the unit sphere constraint. Problem (1.1) can arise from some real-world applications, for example, in the downlink of a multi-user MIMO system [1] and in the sparse Fisher discriminant analysis in pattern recognition [2][3][4][5]. Moreover, it has been pointed out [5] that there are several other problems that can be equivalently transformed to (1.1).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Zhang

2014

Journal of Computational and Applied Mathematics

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h i g h l i g h t s• A self-consistent-field method for optimizing the sum of the Rayleigh quotients is proposed.• Local convergence and global convergence are established under some conditions. • Quadratic convergence is proved for some situations.• Numerical testing shows efficiency, in comparison with other optimization-based methods. MSC:Keywords: Rayleigh quotient The nonlinear eigenvalue problem The self-consistent-field iteration Trust region a b s t r a c tIn this paper, we consider efficient methods for maximizing x ⊤ Bxwhere B, D are symmetric matrices, and W is symmetric and positive definite. This problem can arise in the downlink of a multi-user MIMO system and in the sparse Fisher discriminant analysis in pattern recognition. It is already known that the problem of finding a global maximizer is closely associated with a nonlinear extreme eigenvalue problem. Rather than resorting to some general optimization methods, we introduce a self-consistent-field-like (SCF-like) iteration for directly solving the resulting nonlinear eigenvalue problem. The SCF iteration is widely used for solving the nonlinear eigenvalue problems arising in electronic structure calculations. One attractive feature of the SCF for our problem is that once it converges, the limit point not only satisfies the necessary local optimality conditions automatically, but also, and most importantly, satisfies a global optimality condition, which generally is not achievable in some optimization-based methods. The global convergence and local quadratic convergence rate are proved for certain situations. For the general case, we then discuss a trust-region SCF iteration for stabilizing the SCF iteration, which is of good global convergence behavior. Our preliminary numerical experiments show that these algorithms are more efficient than some optimization-based methods.

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Section: Preliminary Numerical Experimentsmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Zhang

2014

Journal of Computational and Applied Mathematics

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“…not all genes in a gene set are regulated at the expression level in a given physiological condition. Some methods were developed to select subsets of genes simultaneously with the generation of gene set statistics [45][46] , utilize covariance structure in testing 47 , or incorporate regulatory network connectivity 48 . Some of the gene set analysis methods have been previously reviewed and compared [38][39][40][49][50][51][52] .…”

Section: Pathway/metabolite Set Testingmentioning

confidence: 99%

Analyzing LC/MS Metabolic Profiling Data in the Context of Existing Metabolic Networks

Yu¹,

Bai²

2012

CMB

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Metabolic profiling is the unbiased detection and quantification of low molecular-weight metabolites in a living system. It is rapidly developing in biological and translational research, contributing to disease mechanism elucidation, environmental chemical surveillance, biomarker detection, and health outcome prediction. Recent developments in experimental and computational technology allow more and more known metabolites to be detected and quantified from complex samples. As the coverage of the metabolic network improves, it has become feasible to examine metabolic profiling data from a systems perspective, i.e. interpreting the data and performing statistical inference in the context of pathways and genome-scale metabolic networks. Recently a number of methods have been developed in this area, and much improvement in algorithms and databases are still needed. In this review, we survey some methods for the analysis of metabolic profiling data based on metabolic networks.

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“…In recent years, many high-dimensional generalizations of linear discriminant analysis have been proposed (Tibshirani et al, 2002;Trendafilov and Jolliffe, 2007;Clemmensen et al, 2011;Donoho and Jin, 2008;Fan and Fan, 2008;Wu et al, 2008;Shao et al, 2011;Cai and Liu, 2011;Witten and Tibshirani, 2011;Mai et al, 2012;Fan et al, 2012). In the binary case, the discriminant direction is β = Σ −1 (µ 2 − µ 1 ).…”

Section: Introductionmentioning

confidence: 99%

Multiclass Sparse Discriminant Analysis

Zou

Mai²,

Yang

2018

STAT SINICA

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In recent years several sparse linear discriminant analysis methods have been proposed for high-dimensional classification and variable selection. However, most of these proposals focus on binary classification and they are not directly applicable to multiclass classification problems. Some sparse discriminant analysis methods that can handle multiclass classification problems, but their theoretical justifications remain unknown. In this paper, we propose a new multiclass sparse discriminant analysis method that estimates all discriminant directions simultaneously. We show that when applied to the binary case our proposal yields a classification direction that is equivalent to those by two successful binary sparse linear discriminant analysis methods in the literature. Thus, our proposal offers a neat unification of these seemingly unrelated proposals for binary sparse discriminant analysis. Our method can be solved by an efficient algorithm which is implemented in an open R package msda available from CRAN. We offer theoretical justification of our method by establishing a variable selection consistency result and rates of convergence under the ultrahigh dimensionality setting. We Statistica Sinica: Newly accepted Paper (accepted version subject to English editing) Multiclass Sparse Discriminant Analysis 2 further demonstrate the empirical performance of our method on simulated and real data.

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Sparse linear discriminant analysis for simultaneous testing for the significance of a gene set/pathway and gene selection

Abstract: Supplementary data are available at Bioinformatics online.

Cited by 102 publications

References 27 publications

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Analyzing LC/MS Metabolic Profiling Data in the Context of Existing Metabolic Networks

Multiclass Sparse Discriminant Analysis

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