4.5 Article

De-Biased Sparse PCA: Inference for Eigenstructure of Large Covariance Matrices

Journal

IEEE TRANSACTIONS ON INFORMATION THEORY
Volume 67, Issue 4, Pages 2507-2527

Publisher

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2021.3059765

Keywords

Eigenvalues and eigenfunctions; Statistics; Sociology; Covariance matrices; Principal component analysis; Estimation; Loading; Covariance matrix; eigenvectors; eigenvalues; PCA; high-dimensional model; sparsity; Lasso; asymptotic normality; confidence intervals

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Sparse principal component analysis has become widely used for dimensionality reduction, and this paper proposes a methodology for uncertainty quantification with construction of confidence intervals and tests for the principal eigenvector. The novel estimator achieves minimax optimal rates, has a Gaussian limiting distribution, and can be used for hypothesis testing and support recovery of the first eigenvector. The empirical performance of the new estimator is demonstrated on synthetic data and shown to compare favorably with classical PCA in moderately high-dimensional regimes.
Sparse principal component analysis has become one of the most widely used techniques for dimensionality reduction in high-dimensional datasets. While many methods are available for point estimation of eigenstructure in high-dimensional settings, in this paper we propose methodology for uncertainty quantification, such as construction of confidence intervals and tests for the principal eigenvector and the corresponding largest eigenvalue. We base our methodology on an M-estimator with Lasso penalty which achieves minimax optimal rates and is used to construct a de-biased sparse PCA estimator. The novel estimator has a Gaussian limiting distribution and can be used for hypothesis testing or support recovery of the first eigenvector. The empirical performance of the new estimator is demonstrated on synthetic data and we also show that the estimator compares favourably with the classical PCA in moderately high-dimensional regimes.

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