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1. Yang, Ming 1984-. On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing.

Degree: 2012, Texas A&M University

URL: http://hdl.handle.net/1969.1/148351

In this dissertation, we study the partial and generic uniqueness of block term tensor decompositions in signal processing. We present several conditions for generic uniqueness of tensor decompositions of multilinear rank (1, L1, L1), ..., (1, LR, LR) terms. Our proof is based on algebraic geometric methods. Mathematical preliminaries for this dissertation are multilinear algebra, and classical algebraic geometry.
In geometric language, we prove that the joins of relevant subspace varieties are not tangentially weakly defective. We also give conditions for partial uniqueness of block term tensor decompositions by proving that the joins of relevant subspace varieties are not defective. The main result is the following. For a tensor Y belong to the tensor product of three complex vector spaces of dimensions I, J, K, we assume that L1, L2, ..., LR is from small to large, K is bigger or equal to J, and J is strictly bigger than LR. If the dimension of ambient space is strictly less than IJK, then for general tensors among those admitting block term tensor decomposition, the block term tensor decomposition is partially unique under the condition that the binomial coefficient indexed by J and LR is bigger or equal to R, and I is bigger or equal to 2; it has infinitely many expressions under the condition IJK is strictly less than the sum from L_{1}^{2} to L_{R}^{2}; it is essentially unique under any of the following there conditions: (i) I is bigger or equal to 2, J, K is bigger or equal to the sum from L1 to LR (ii) R is 2, I is bigger or equal to 2 (iii) I is bigger or equal to R, K is bigger or equal to the sum from L1 to LR, J is bigger or equal to 2LR, the binomial coefficient indexed by J and LR is bigger or equal to R.
*Advisors/Committee Members: Landsberg, Joseph (advisor), Becker, Melanie (committee member), Robles, Colleen, Sottile, Frank, Zelenkko, Igor (committee member).*

Subjects/Keywords: tangentially weakly defectivity; subspace varieties

…Definition 1 in [22]) *Subspace* *varieties*, denoted by Subk1 ,...,kn (A1… …Infinitesimal study of *subspace* *varieties*
For the benefit of the reader, we recall the following… …*varieties* in Grassmannians.
Definition 2.3.6. For any k ≤ n − m and any *subspace* W ⊆ Cn of… …space V , its dual space is denoted V ∗ . If A ⊂ V is a *subspace*,
A⊥ ⊂ V ∗ is its annihilator… …When U ⊂ V is a linear *subspace*
one also says that PU ⊂ PV is a linear *subspace*. Just as in…

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APA (6^{th} Edition):

Yang, M. 1. (2012). On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing. (Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/148351

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Yang, Ming 1984-. “On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing.” 2012. Thesis, Texas A&M University. Accessed October 16, 2019. http://hdl.handle.net/1969.1/148351.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Yang, Ming 1984-. “On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing.” 2012. Web. 16 Oct 2019.

Vancouver:

Yang M1. On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing. [Internet] [Thesis]. Texas A&M University; 2012. [cited 2019 Oct 16]. Available from: http://hdl.handle.net/1969.1/148351.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Yang M1. On Partial and Generic Uniqueness of Block Term Tensor Decomposition in Signal Processing. [Thesis]. Texas A&M University; 2012. Available from: http://hdl.handle.net/1969.1/148351

Not specified: Masters Thesis or Doctoral Dissertation