Beyond matrices: theory, methods, and applications of higher-order tensors
Friday, January 26, 2018 - 3:30pm
Recently, tensors of order 3 or greater, known as higher-order tensors, have attracted increased attention in modern scientific contexts including genomics, proteomics, and brain imaging. A common paradigm in tensor-based algorithms advocates unfolding tensors into matrices and applying classical matrix methods. In this talk, I will consider all possible unfoldings of a tensor into lower order tensors and present general inequalities between their operator norms. I will then present a new tensor decomposition algorithm built on these inequalities and Kruskal’s uniqueness theorem. This tensor decomposition algorithm provably handles a greater level of noise compared to previous methods while achieving high estimation accuracy. A perturbation analysis is provided, establishing a higher-order analogue of Wedin's perturbation theorem. In light of these theoretical successes, I will discuss a novel tensor approach to three-way clustering for multi-tissue multi-individual gene expression studies. Finally, I'll show that the tensor method deciphers multi-way transcriptome specificity in finer detail than previously possible via our analysis of the GTEx RNA-seq data.
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