This is a minisymposium on the development of numerical algorithms for multilinear algebra — a topic that has become increasingly important in science and engineering, through both the creation of new scientific computing models and the analysis of data with nonlinear structures. We broadly define Numerical Multilinear Algebra as the study and use of tensors/multilinear algebra, symmetric tensors/symmetric algebra, alternating tensors/exterior algebra, spinors/Clifford algebra in computational mathematics. More specifically, this minisymposium will focus on numerical computations involving various multilinear objects and the ubiquity of multilinearity in scientific and engineering applications.
The minisymposium will be held as a part of the 2008 SIAM Annual Meeting at the Town and Country Resort and Convention Center in San Diego, CA. Check out our minisymposium of the same title last year at the 6th International Congress on Industrial and Applied Mathematics (ICIAM 2007).
The official program is now online. Our minisymposium (MS 102) will take place on Friday, July 11, 10:30am–12:30pm in the room Eaton.
Shmuel Friedland, University of Illinois, Chicago
Abstract: In many applications, it is of interest to approximate data, given by m by n matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time consuming for very large m and n. We present here an optimal least squares algorithm for computing a k-rank approximation to the data consisting of m by n matrix A by reading a number of rows and columns of A. This algorithm can be applied to nonnegative matrices and to tensors.
Guangliang Chen and Gilad Lerman, University of Minnesota, Twin Cities
Abstract: The polar curvature describes the d-dimensional flatness of d + 2 points in a Euclidean space. We utilize the polar curvature tensor to form a multi-way spectral clustering algorithm for solving the problem of segmenting data sampled from a mixture of sufficiently flat surfaces. Using tensorial algebra and matrix perturbation theory, we prove that, with high probability, our proposed algorithm segments the different underlying surfaces well. We exemplify application of the algorithm to several real-world problems.
Gregory Beylkin, University of Colorado, Boulder
Abstract: An efficient representation of multi-variable functions via (a separable) sums of exponentials has many applications in numerical analysis and signal processing. Obtained for a finite but arbitrary accuracy in the case of single variable, such representations typically have significantly fewer terms than corresponding Fourier expansions. We present examples of these approximations, discuss their applications and describe current work towards multidimensional extensions of our approach.
Bernard Mourrain, INRIA Sophia Antipolis, France
Abstract: In 1886, J.J. Sylvester proposed a method to decompose a binary form into a minimal sum of powers of linear forms, based on rank and kernel computation of specific Hankel matrices. In this talk, we will revisit this approach for multivariate polynomials from a dual point of view, show how it is related to normal form computation and to truncated moment problems and describe an algorithm to compute such a decomposition.
For further information on this event, please email Lek-Heng Lim at lekheng(at)math.berkeley.edu.