Sparse principal component analysis and iterative thresholding

Ma, Zongming

doi:10.1214/13-aos1097

Cited by 263 publications

(289 citation statements)

References 31 publications

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“…Birnbaum et al (2012) also proposed an estimation procedure named ASPCA which is based on a two-stage coordinate selection scheme. Both these estimators are rateoptimal, so long as the diagonal thresholding scheme is consistent (see Ma, 2013;Paul and Johnstone, 2012). A different two-stage estimation scheme, based on a regression framework, has been proposed by .…”

Section: Covariance Estimation Using Sparse Factor Modelsmentioning

confidence: 99%

“…A different two-stage estimation scheme, based on a regression framework, has been proposed by . Incidentally, in order to establish upper bounds on rates of convergence of the estimators, Amini and Wainwright (2008), Ma (2013) and Paul and Johnstone (2012) all utilize results on the concentration of the extreme singular values of a rectangular matrix with i.i.d. standard Gaussian entries, which translates into finite sample probabilistic bounds on the extreme eigenvalues of a Wishart matrix.…”

Section: Covariance Estimation Using Sparse Factor Modelsmentioning

confidence: 99%

See 1 more Smart Citation

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

166

115

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b s t r a c tWe give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT.

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Section: Covariance Estimation Using Sparse Factor Modelsmentioning

confidence: 99%

Section: Covariance Estimation Using Sparse Factor Modelsmentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

166

115

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show abstract

“…The separations between the common factors and idiosyncratic components are carried out by the low-rank plus sparsity decomposition. See, for example, Cai et al (2013); Candès and Recht (2009) ;Fan et al (2013); Koltchinskii et al (2011);Ma (2013); Negahban and Wainwright (2011).…”

Section: Introductionmentioning

confidence: 99%

Projected Principal Component Analysis in Factor Models

2014

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This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employees principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semi-parametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates' effects on the factor loadings are further modeled by the additive model via sieve approximations. By using the newly proposed Projected-PCA, the rates of convergence of the smooth factor loading matrices are obtained, which are much faster than those of the conventional factor analysis. The convergence is achieved even when the sample size is finite and is particularly appealing in the high-dimension-low-sample-size situation. This leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings. The proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index.

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“…The basic idea is to produce the optimal simplification of a multi variable system in which the p variable values x 1 , x 2 ,…, x p described in the data sheet of the original system re-adjust and combine, and extract m (m ≤ p) synthesize variable f 1 , f 2 , …, f m , which can maximize the overlap of the information of data object described in the original system. Dimensionality reduction and the simplification goal can then be obtained [12][13][14]. They can be represented: (10) Where m f is the principal component, and m ≤ p. Generally, the cumulative contribution of m principal components is 85%, so in keeping the most information, the goal of dimensionality reduction can be achieve.…”

Section: Principal Component Analysismentioning

confidence: 99%