The complexity of bilinear maps (equivalently, of $3$-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for $3$-mode tensors, this correspondence breaks down for $d \geq 4$ modes. As a result, the complexity of $d$-mode tensors for larger fixed $d$ remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen's $2ω/3$ bound on the asymptotic rank exponent of $3$-mode tensors. Our proof directly generalizes to an upper bound of $(d-1)ω/3$ for $d$-mode tensors. Using refined techniques available only for $d\geq 4$ modes, we improve this bound beyond the current state of the art for $ω$. We also obtain a bound of $d/2+1$ on the asymptotic exponent of circuit complexity of generic $d$-mode tensors and optimized bounds for $d \in \{4,5\}$. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of $T$ and $U$ imply low complexity of their Kronecker product $T \otimes U$. While this crucially holds for rank (and thus for circuit complexity in $3$ modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for $d=8$ modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors $T_H$, as employed by Christandl and Zuiddam ({\em Comput.~Complexity}~28~(2019)~27--56) and [...]

翻译：双线性映射（等价于三模态张量）的复杂度已被广泛研究，其中最著名的是矩阵乘法背景下的研究。虽然对于三模态张量，电路复杂度与张量秩在渐近意义下一致，但这种对应关系在模态数 $d \geq 4$ 时失效。因此，尽管具有重要应用价值（例如在精细复杂度理论中），对于更大固定 $d$ 值的 $d$ 模态张量复杂度仍缺乏深入理解。本文探索了这一中间区域。首先，我们给出 Strassen 关于三模态张量渐近秩指数 $2ω/3$ 上界的"图论"证明。该证明可直接推广至 $d$ 模态张量的 $(d-1)ω/3$ 上界。利用仅适用于 $d\geq 4$ 模态的精细技术，我们改进了当前关于 $ω$ 的最佳上界。同时，我们获得了通用 $d$ 模态张量电路复杂度渐近指数的 $d/2+1$ 上界，并针对 $d \in \{4,5\}$ 给出了优化边界。据我们所知，张量的渐近电路复杂度（而非秩）此前尚未被研究。为建立稳健的理论体系，我们首先探讨 $T$ 和 $U$ 的低复杂度是否意味着其 Kronecker 积 $T \otimes U$ 也具有低复杂度。虽然这一性质对秩（以及三模态下的电路复杂度）至关重要，但我们证明精细复杂度理论中的假设排除了多模态张量电路复杂度具有此类次可乘性的可能性。特别地，在超团猜想成立的条件下，这种失效现象在 $d=8$ 模态时即已出现。尽管如此，我们仍可保留受限的次可乘性概念。从技术角度看，我们的证明大量使用了 Christandl 与 Zuiddam（《计算复杂性》28卷（2019年）27-56页）及后续研究采用的图张量 $T_H$ [...]