The Annals of Statistics

Statistical inference based on robust low-rank data matrix approximation

Xingdong Feng and Xuming He

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The singular value decomposition is widely used to approximate data matrices with lower rank matrices. Feng and He [Ann. Appl. Stat. 3 (2009) 1634–1654] developed tests on dimensionality of the mean structure of a data matrix based on the singular value decomposition. However, the first singular values and vectors can be driven by a small number of outlying measurements. In this paper, we consider a robust alternative that moderates the effect of outliers in low-rank approximations. Under the assumption of random row effects, we provide the asymptotic representations of the robust low-rank approximation. These representations may be used in testing the adequacy of a low-rank approximation. We use oligonucleotide gene microarray data to demonstrate how robust singular value decomposition compares with the its traditional counterparts. Examples show that the robust methods often lead to a more meaningful assessment of the dimensionality of gene intensity data matrices.

Article information

Ann. Statist., Volume 42, Number 1 (2014), 190-210.

First available in Project Euclid: 18 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing 62F35: Robustness and adaptive procedures
Secondary: 62F05: Asymptotic properties of tests 62F10: Point estimation 62F12: Asymptotic properties of estimators

Hypothesis testing M estimator singular value decomposition trimmed least squares


Feng, Xingdong; He, Xuming. Statistical inference based on robust low-rank data matrix approximation. Ann. Statist. 42 (2014), no. 1, 190--210. doi:10.1214/13-AOS1186.

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Supplemental materials

  • Supplementary material: Additional details of case study and technical proofs. We provide details of the case study in Section 4.3 and complete the proofs of technical lemmas, as well as Theorems 3.1–3.2 and 4.2 of this paper.