Electronic Journal of Statistics

Reconstruction of a high-dimensional low-rank matrix

Kazuyoshi Yata and Makoto Aoshima

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We consider the problem of recovering a low-rank signal matrix in high-dimensional situations. The main issue is how to estimate the signal matrix in the presence of huge noise. We introduce the power spiked model to describe the structure of singular values of a huge data matrix. We first consider the conventional PCA to recover the signal matrix and show that the estimation of the signal matrix holds consistency properties under severe conditions. The conventional PCA is heavily subjected to the noise. In order to reduce the noise we apply the noise-reduction (NR) methodology and propose a new estimation of the signal matrix. We show that the proposed estimation by the NR method holds the consistency properties under mild conditions and improves the error rate of the conventional PCA effectively. Finally, we demonstrate the reconstruction procedures by using a microarray data set.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 895-917.

Received: October 2015
First available in Project Euclid: 8 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62F12: Asymptotic properties of estimators

Eigenstructure HDLSS noise-reduction methodology PCA singular value decomposition


Yata, Kazuyoshi; Aoshima, Makoto. Reconstruction of a high-dimensional low-rank matrix. Electron. J. Statist. 10 (2016), no. 1, 895--917. doi:10.1214/16-EJS1128. https://projecteuclid.org/euclid.ejs/1460141647

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