Numerical multilinear algebra is an emerging topic in computational
mathematics. It may be broadly described as the numerical and
algorithmic studies of tensors and multilinear algebra, symmetric
tensors and symmetric algebra, alternating tensors and exterior algebra,
spinors and Clifford algebra --- objects where multilinearity plays a
central role. An order-k tensor may either be regarded as (1) a
k-dimensional array of real/complex numbers on which algebraic
operations generalizing analogous operations on matrices are defined, or
(2) a linear combination of outer products of vectors. A matrix is then
synonymous with a tensor of order 2. Special types of tensors such as
symmetric and alternating tensors and Kronecker products of operators
may also be defined.

We will discuss how numerical multilinear algebra arises in both
discriminative and generative models in machine learning: tensors in
various multilinear statistical models (generalization of vector space
models), symmetric tensors in independent component analysis,
nonnegative tensors in graphical models (ie. Bayesian networks). We will
also introduce a multilinear spectral theory and show how the
eigenvalues of symmetric tensors may be used to obtain basic results in
Spectral Hypergraph Theory. We will illustrate our talk with selected
applications in bioinformatics, computer vision, signal processing, and
spectroscopy.