Tensor rank : some lower and upper bounds

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) </-IGId/2 In the case that d = 3, we are able to show that rank(T3G) </- O(IGIw/2) </- O(IGIV1.19), where w is the exponent of matrix multiplication. The next upper bound uses interpolation and only works for abelian G, showing that over any field F that rank F(TdG) </- 0(IGI)1+1gd-1 IGI). In either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank. We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n]d --> F that have tensor rank at most dn but have monotone tensor rank exactly nd-1 . This is a nearly optimal separation.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2011

Author dc:creator

• Forbes, Michael Andrew

Advisor dc:contributor.advisor

• Scott Aaronson.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/66029

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/66029