Tensor rank

Thursday, January 28, 2016 - 10:15am - 11:05am
Shuzhong Zhang (University of Minnesota, Twin Cities)
This talk presents two parts of results related to tensor computations. The first part is on a new matricization approach, resulting in lower and upper approximations for the CP-rank. In this part, theoretical properties of the new ranks (to be called the M-ranks) will be discussed, with applications to solve the low CP-rank tensor completion problem. The second part establishes the equivalence between a tensor and its Tucker core, in terms of the CP-rank, the Schatten norms, and the Z-eigenvalues of tensors.
Wednesday, January 27, 2016 - 10:15am - 11:05am
Lek-Heng Lim (University of Chicago)
We show that in many instances, at the heart of a problem in numerical computation sits a special 3-tensor, the structure tensor of the problem that uniquely determines its underlying algebraic structure. Any decomposition of the structure tensor into rank-1 terms gives an explicit algorithm for solving the problem. The rank of the structure tensor measures the speed of the fastest possible algorithm for the problem, whereas the nuclear and spectral norms quantify the numerical stability of the stablest algorithm for the problem.
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