Tensor rank : some lower and upper bounds
Name
752141816-MIT.pdf
Description
Full printable version
Size
2.75 MB
Format
Adobe PDF
Checksum (MD5)
940d066ad7bc622eb7d6069fbf9cfcbc
Author(s)
Forbes, Michael Andrew
Advisor(s)
Scott Aaronson.
Date Issued
2011
Publisher
Massachusetts Institute of Technology
Abstract
The results of Strassen [25] and Raz [19] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n]d --> F with rank at least 2n td/ 2j + n - [theta](d Ig n). This improves the lower-order terms in known lower bounds for any odd d >/- 3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by applying known counting lower bounds. that there exist order-3 permutation tensors with super-linear rank as well as order-d permutation tensors with high rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor TdG : Gd --> F, by TdG(g1 .....gd) = 1 iff g1 ...gd = 1G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over "large" fields F, showing (among other things) that rankF(TdG) F that have tensor rank at most dn but have monotone tensor rank exactly nd-1 . This is a nearly optimal separation.
Description
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 59-61).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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