Tensor Rank: Some Lower and Upper Bounds

• Authors: Boris Alexeev;Michael A. Forbes;Jacob Tsimerman
• Venue: CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
• Year: 2011

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.