Tensor rank: Some lower and upper bounds

The results of Strassen [1] and Raz [2] 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^[d/2] + n - Θ(d lg 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 orderd 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 → F, by T _G^d(g₁,⋯,g_d) = 1 iff g ₁ ··· 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 F, showing (among other things) that rank_F(T_G^d) ≤ |G|^d/2. In the case that d = 3, we are able to show that rank_F(T_G ^d) ≤ O|G|^ω/2) ≤ O|G|^1.19), where ω 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(T_G^d) ≤ O(|G|^{1+lg d lgd-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 → F that have tensor rank at most dn but have monotone tensor rank exactly n^d-1. This is a nearly optimal separation.