Supplemental material for S. P. Ponnapalli, M. A. Saunders, C. F. Van Loan and O. Alter, "A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms," Public Library of Science (PLoS) One 6 (12), article e28072 (December 2011); doi: 10.1371/journal.pone.0028072.
Mention: Among the top 10% most cited Public Library of Science (PLoS) One articles as of 2017, PLoS One (June 30, 2017).
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Abstract:
The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices D_i ∈ ℜ^m_i×n, each with full column rank. Each matrix is exactly factored as D_i = U_iΣ_iV^T, where V, identical in all factorizations, is obtained from the eigensystem SV = VΛ of the arithmetic mean S of all pairwise quotients A_i A_j^-1 of the matrices A_i = D_i^TD_i, i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λ_k≥1. Equality holds if and only if the corresponding eigenvector v_k is a right basis vector of equal significance in all matrices D_i and D_j, that is σ_i,k/σ_j,k = 1 for all i and j, and the corresponding left basis vector u_i,k is orthogonal to all other vectors in U_i for all i. The eigenvalues λ_k=1, therefore, define the "common HO GSVD subspace." We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified.

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• Dataset S1: S_pombe.txt

S. pombe global mRNA expression.
A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of m₁=3167 S. pombe gene clones at n=17 time points during about two cell-cycle periods from Rustici et al. and the cell-cycle classifications from Oliva et al.

• Dataset S2: S_cerevisiae.txt

S. cerevisiae global mRNA expression.
A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of m₂=4772 S. cerevisiae open reading frames (ORFs), or genes, at n=17 time points during about two cell-cycle periods, including cell-cycle classifications, from Spellman et al.

• Dataset S3: Human.txt

Human global mRNA expression.
A tab-delimited text format file, readable by both Mathematica and Microsoft Excel, reproducing the relative mRNA expression levels of m₃=13,068 human gene clones at n=17 time points during about two cell-cycle periods, including cell-cycle classifications, from Whitfield et al.