张量秩：一些上下界
Tensor Rank: Some Lower and Upper Bounds
摘要：
Strassen和Raz的结果表明，足够好的张量秩下界对代数电路/公式下界有影响。我们探讨张量秩的下限和上限，专注于显式张量。对于奇数d，我们构造秩至少为2n^（floor（d/2））+n-Theta（d log n）的场独立显式0/1张量T：[n]^d->F。这匹配（在F_2上）或改进（所有其他字段）已知的d=3的下限，并改进（在任何字段上）奇数d>3。我们还探讨了置换矩阵的推广，我们表示置换张量。通过计数证明了存在一个超线性秩的三阶置换张量。我们还探讨了一个自然类的置换张量，我们称之为组张量。对于任何群G，我们定义群张量T_G^d：G^d->F，通过T_G^d（g_1，...，g_d）=1当且仅当g_1 x. x g_d=1_G。我们给这些张量的秩的两个上界。第一种方法使用表示理论，并在大域F上工作，表明（除其他事项外）rank_F（T_G^d）<= |G| ^（d/2）。我们还表明，如果这个上界是紧的，那么超线性张量秩下界将遵循。第二个上界使用插值，并且只对阿贝尔G有效，表明在任何域F上rank_F（T_G^d）<= O（|G| ^（1+log d）log^（d-1）|G|）。在任何一种情况下，这表明许多置换张量都远离最大秩，这与矩阵的情况非常不同，因此消除了许多高张量秩的自然候选者。我们还探讨了单调张量秩。我们给出显式0/1张量T：[n]^d->F，其张量秩至多为dn，但单调张量秩恰好为n^（d-1）。这是一种近乎最佳的分离。
英文摘要：
The results of Strassen and 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->F with rank at least 2n^(floor(d/2))+n-Theta(d log n). This matches (over F_2) or improves (all other fields) known lower bounds for d=3 and improves (over any field) for odd d>3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear 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(g_1,...,g_d)=1 iff g_1 x ... x 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). We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second 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+log d)log^(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->F that have tensor rank at most dn but have monotone tensor rank exactly n^(d-1). This is a nearly optimal separation.
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