Georgia Institute of Technology College of Sciences

Numerical linear algebra is often regarded as a workhorse of scientific and engineering computing. Computational problems arising from optimization, partial differential equation, statistical estimation, etc, are usually reduced to one or more standard problems involving matrices: linear systems, least squares, eigenvectors/singular vectors, low-rank approximation, matrix nearness, etc. The idea of developing numerical algorithms for multilinear algebra is naturally appealing -- if similar problems for tensors of higher order (represented as hypermatrices) may be solved effectively, then one would have substantially enlarged the arsenal of fundamental tools in numerical computations. We will see that higher order tensors are indeed ubiquitous in applications; for multivariate or non-Gaussian phenomena, they are usually inevitable. However the path from linear to multilinear is not straightforward. We will discuss the theoretical and computational difficulties as well as ways to avoid these, drawing insights from a variety of subjects ranging from algebraic geometry to compressed sensing. We will illustrate the utility of such techniques with our work in cancer metabolomics, EEG and fMRI neuroimaging, financial modeling, and multiarray signal processing.