Colloquium October 3
Date: Tuesday October 3
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Carla D. (Moravitz) Martin of James Madison
Title: The Rank of a Tensor
Abstract
Determining the rank of a matrix is straight-forward using the
Singular Value Decomposition (SVD); the number of non-zero singular
values in the decomposition equals the rank of a matrix. As computing
power increases, many more problems in engineering and data analysis
involve computation with tensors, or multi-way data arrays. Most
applications involve computing a decomposition of a tensor into a
linear combination of rank-1 tensors. Ideally, the decomposition
involves a minimal number of terms, i.e., computation of the rank of
the tensor. The rank of a general tensor is quite complicated. In
fact, computing the rank of an arbitrary tensor is an open problem. In
this talk, I provide insight into the connection between tensor rank
and eigenvalues of certain matrices.
I will begin by illustrating some major differences between matrix
rank and tensor rank. The main contribution is an explicit algorithm
to compute the rank of a small subclass of tensors. For clarity, we
begin with $2\times 2\times 2$ tensors and extend it to $n\times
n\times 2$ tensors. These results provide insight into the complexity
of tensor rank. Perturbation results will be presented and we
conclude with some open problems related to tensor rank.
Return to