Due to the exponential growth of the dimension of the space of tensors V_(1)⊗• • •⊗V_(n), any naive method of representing these tensors is intractable on a computer. In practice, we consider feasible subspaces (subvarieties) which are defined to reduce the storage cost and the computational complexity. In this thesis, we study two such types of subvarieties: the third secant variety of the product of n projective spaces, and tensor network states.
For the third secant variety of the product of n projective spaces, we determine set-theoretic defining equations, and give an upper bound of the degrees of these equations.
For tensor network states, we answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/151240 |
| Date | 16 December 2013 |
| Creators | Qi, Yang |
| Contributors | Landsberg, Joseph, Sottile, Frank, Lima-Filho, Paulo, Robles, Colleen, Becker, Katrin |
| Source Sets | Texas A and M University |
| Language | English |
| Detected Language | English |
| Type | Thesis, text |
| Format | application/pdf |
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