Tuesday, February 20, 2007
4:15 pm
5130 Upson Hall

Computer Science
Spring 2007

Lek-Heng Lim
Stanford University

Numerical Multilinear Algebra in Data Analysis

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.