Tensor generalizations of the singular value decomposition for integrative analysis of large-scale molecular biological data
Year: 2007
Type: dissertation
Abstract: The structure of large-scale molecular biological data is often of an order higher than \nthat of a matrix, especially when integrating data from different studies. Flattened \ninto a matrix format, much of the information in the data is lost. I describe the \nuse of higher-order generalizations of singular value decomposition (SVD) - both \nthe higher-order singular value decomposition (HOSVD) and Parallel Factorization \n(PARAFAC) - in transforming tensors into simplified spaces. I apply these transformations \nto a series of DNA microarray datasets from different studies tabulated \nin a tensor of genes × time × conditions, specifically an integration of genome-scale \nmRNA expression data from three yeast-cell cycle time courses. One of the time \ncourses was under exposure to the oxidative stress agent hydrogen peroxide (HP); \nanother was exposed to menadione (MD) and the third was unstressed[45]. \nThe HOSVD transforms the tensor to a “core tensor” of “eigenarrays” × “timeeigengenes” \n× “condition-eigengenes,” where the eigenarrays, time-eigengenes and \ncondition-eigengenes are unique orthonormal superpositions of the genes, times and \nconditions, respectively. This HOSVD, also known as N-mode SVD, formulates the \ntensor as a linear superposition of all possible outer products of an eigenarray, a timeeigengene \nand a condition-eigengene, i.e., rank-1 “subtensors,” the superposition \ncoefficients of which are tabulated in the core tensor. Each coefficient indicates the \nsignificance of the corresponding subtensor in terms of the overall information it \ncaptures in the data. PARAFAC reformulates the same data tensor into a sum of \nrank-1 tensor of F elements that best approximate the data tensor in a least square \nsense. \nI show that significant rank-1 subtensors can be associated with independent biological \nprocesses, which are manifested in the data tensor. Subtensors of the HOSVD \ncapture the subprocesses: stress response, pheromone response and developmental \nstage. The data suggests that the conserved genes YKU70, MRE11, AIF1 and \nZWF1, as well as the genes involved in the processes of retrotransposition, apoptosis \nand the oxidative pentose phosphate cycle may play significant, yet previously unrecognized, \nroles in the differential effects of HP and MD on cell cycle progression. \nSubtensors of PARAFAC capture the same biological processes as the 2 most significant \nHOSVD subtensors. A genome-wide correlation between DNA replication \nand initiation of RNA transcription, which is equivalent to a recently discovered \ncorrelation and might be due to a previously unknown mechanism of regulation, is \nindependently uncovered.
Author: Lars Gustaf Omberg
Language: English
Subfield: Computational Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2285542516