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Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties

McClatchey, Nathaniel J.
http://orcid.org/0000-0002-4961-1457
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520508234977924

Abstract Details

2018, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
This dissertation addresses several problems related to low-rank approximation of tensors. Low-rank approximation of tensors is plagued by slow convergence of the sequences produced by popular algorithms such as Alternating Least Squares (ALS), by ill-posed approximation problems which cause divergent sequences, and by poor understanding of the nature of low-rank tensors. Though ALS may produce slowly-converging sequences, ALS remains popular due to its simplicity, its robustness, and the low computational cost for each iteration. I apply insights from Successive Over-Relaxation (SOR) to ALS, and develop a novel adaptive method based on the resulting Successive Over-Relaxed Least Squares (SOR-ALS) method. Numerical experiments indicate that the adaptive method converges more rapidly than does the original ALS algorithm in almost all cases. Moreover, the adaptive method is as robust as ALS, is only slightly more complicated than ALS, and each iteration requires little computation beyond that of an iteration of ALS. Divergent sequences in tensor approximation may be studied by examining their images under some map. In particular, such sequences may be re-scaled so that they become bounded, provided that the objective function is altered correspondingly. I examine the behavior of sequences produced when optimizing bounded multivariate rational functions. The resulting theorems provide insight into the behavior of certain divergent sequences. Finally, to improve understanding of the nature of low-rank tensors, I examine connectedness properties of spaces of low-rank tensors. I demonstrate that spaces of unit tensors of bounded rank are path-connected if the space of unit vectors in at least one of the factor spaces is path-connected, and that spaces of unit separable tensors are simply-connected if the unit vectors are simply-connected in every factor space. Moreover, I partially address simple connectedness for unit tensors of higher rank.
Martin Mohlenkamp (Advisor)
Todd Young (Committee Member)
Winfried Just (Committee Member)
David Juedes (Committee Member)
135 p.
Mathematics
tensor; CP decomposition; alternating least squares; successive over-relaxation; global convergence; Lojasiewicz inequality; connectedness

Recommended Citations

Citations

  • McClatchey, N. J. (2018). Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520508234977924

    APA Style (7th edition)

  • McClatchey, Nathaniel. Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties. 2018. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520508234977924.

    MLA Style (8th edition)

  • McClatchey, Nathaniel. "Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties." Doctoral dissertation, Ohio University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1520508234977924

    Chicago Manual of Style (17th edition)

ohiou1520508234977924
412
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This open access ETD is published by Ohio University and OhioLINK.