TY - JOUR
T1 - A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms
A1 - Ponnapalli, Sri Priya
A1 - Saunders, Michael A.
A1 - Van Loan, Charles F.
A1 - Alter, Orly
Y1 - 2011/12/22
N2 - 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 , each with full column rank. Each matrix is exactly factored as D_{i} = U_{i}Σ_{i}V^{T}, where V, identical in all factorizations, is obtained from the eigensystem SV = VΛ of the arithmetic mean S of all pairwise quotients of the matrices , 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.

JF - PLoS ONE
JA - PLoS ONE
VL - 6
IS - 12
UR - http://dx.doi.org/10.1371%2Fjournal.pone.0028072
SP - e28072
EP -
PB - Public Library of Science
M3 - doi:10.1371/journal.pone.0028072
ER -