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2011 26th Annual IEEE Conference on Computational Complexity
Tensor Rank: Some Lower and Upper Bounds
San Jose, California USA
June 08June 11
ISBN: 9780769544113
ASCII Text  x  
Boris Alexeev, Michael A. Forbes, Jacob Tsimerman, "Tensor Rank: Some Lower and Upper Bounds," 2012 IEEE 27th Conference on Computational Complexity, pp. 283291, 2011 26th Annual IEEE Conference on Computational Complexity, 2011.  
BibTex  x  
@article{ 10.1109/CCC.2011.28, author = {Boris Alexeev and Michael A. Forbes and Jacob Tsimerman}, title = {Tensor Rank: Some Lower and Upper Bounds}, journal ={2012 IEEE 27th Conference on Computational Complexity}, volume = {0}, year = {2011}, issn = {10930159}, pages = {283291}, doi = {http://doi.ieeecomputersociety.org/10.1109/CCC.2011.28}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  CONF JO  2012 IEEE 27th Conference on Computational Complexity TI  Tensor Rank: Some Lower and Upper Bounds SN  10930159 SP283 EP291 A1  Boris Alexeev, A1  Michael A. Forbes, A1  Jacob Tsimerman, PY  2011 KW  tensor rank KW  algebraic complexity VL  0 JA  2012 IEEE 27th Conference on Computational Complexity ER   
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2011.28
The results of Strassen~\cite{strassentensor} 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 fieldindependent 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 lowerorder 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 order3 permutation tensors with superlinear 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^{d1}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^{d1}. This is a nearly optimal separation.
Index Terms:
tensor rank, algebraic complexity
Citation:
Boris Alexeev, Michael A. Forbes, Jacob Tsimerman, "Tensor Rank: Some Lower and Upper Bounds," ccc, pp.283291, 2011 26th Annual IEEE Conference on Computational Complexity, 2011
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