NDLI: Tensor Rank: Some Lower and Upper Bounds

Content Provider	IEEE Xplore Digital Library
Author	Alexeev, B. Forbes, M.A. Tsimerman, J.
Copyright Year	2011
Abstract	The results of Strassen~\cite{strassen-tensor} and Raz~\cite{raz} 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\to\mathbb{F} with rank at least 2n^{\lfloor d/2\rfloor}+n-\Theta(d\lg n). This improves the lower-order terms in known lower bounds for any odd d\ge 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 T_G^d:G^d\to\mathbb{F}, by T_G^d(g_1,\ldots,g_d)=1$ iff $g_1\cdots g_d=1_G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over ``large'' fields $\mathbb{F}, showing (among other things) that \rank_\mathbb{F}(T_G^d)\le \|G\|^{d/2}. In the case that d=3, we are able to show that \rank_\mathbb{F}(T_G^3)\le O(\|G\|^{\omega/2})\le O(\|G\|^{1.19}), where $\omega$ is the exponent of matrix multiplication. The next upper bound uses interpolation and only works for abelian G, showing that over any field \mathbb{F}$ that $\rank_\mathbb{F}(T_G^d)\le O(\|G\|^{1+\lg d}\lg^{d-1}\|G\|). 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\to\mathbb{F} that have tensor rank at most $dn$ but have monotone tensor rank exactly n^{d-1}. This is a nearly optimal separation.
Starting Page	283
Ending Page	291
File Size	287919
Page Count	9
File Format	PDF
ISBN	9781457701795
ISSN	10930159
DOI	10.1109/CCC.2011.28
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2011-06-08
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Tensile stress Polynomials Upper bound Integrated circuit modeling Computational modeling Complexity theory USA Councils algebraic complexity tensor rank
Content Type	Text
Resource Type	Article

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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