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. 2016;111(514):621-633.
doi: 10.1080/01621459.2015.1021005. Epub 2016 Aug 18.

Structured Matrix Completion with Applications to Genomic Data Integration

Tianxi Cai  1 , T Tony Cai  2 , Anru Zhang  3
Affiliations

Structured Matrix Completion with Applications to Genomic Data Integration

Tianxi Cai et al. J Am Stat Assoc. 2016.

Abstract

Matrix completion has attracted significant recent attention in many fields including statistics, applied mathematics and electrical engineering. Current literature on matrix completion focuses primarily on independent sampling models under which the individual observed entries are sampled independently. Motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness by design. Specifically, our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low-rank matrix are observed. We provide theoretical justification for the proposed SMC method and derive lower bound for the estimation errors, which together establish the optimal rate of recovery over certain classes of approximately low-rank matrices. Simulation studies show that the method performs well in finite sample under a variety of configurations. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival.

Keywords: Constrained minimization; genomic data integration; low-rank matrix; matrix completion; singular value decomposition; structured matrix completion.

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Figures

Figure 1
Figure 1
Illustrative example with A ∈ ℝ30×30, m1 = m2 = 10. (A darker block corresponds to larger magnitude.)
Figure 2
Figure 2
Searching for the appropriate position to truncate from r̂ = 10 to 1.
Figure 3
Figure 3
Spectral norm loss (left panel) and Frobenius norm loss (right panel) when there is a gap between σr(A) and σr+1(A). The singular value values of A are given by (21), p1 = p2 = 1000, and m1 = m2 = 50.
Figure 4
Figure 4
Spectral norm loss (left panel) and Frobenius norm loss (right panel) as the thresholding constant c varies. The singular values of A are {j−α, j = 1, 2, …} with α varying from 0.3 to 2, p1 = p2 = 1000, and m1 = m2 = 50.
Figure 5
Figure 5
Spectral and Frobenius norm losses with column/row thresholding. The singular values of A are {j−1, j = 1, 2, …}, p1 = 300, p2 = 3000, and m1, m2 = 10, …, 150.
Figure 6
Figure 6
Comparison of the proposed SMC method with the NNM method with 5-cross-validation for the settings with singular values of A being {j−α, j = 1, 2, …} for α ranging from 0.6 to 2, p1 = p2 = 500, and m1 = m2 = 50 or 100.
Figure 7
Figure 7
Imputation scheme for integrating multiple OC genomic studies.
See this image and copyright information in PMC

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