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  "path": "/abs/2602.11975v1",
  "publishedAt": "2026-02-13T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Cornelius Brand",
    "Radu Curticapean",
    "Petteri Kaski",
    "Baitian Li",
    "Ian Orzel",
    "Tim Seppelt",
    "Jiaheng Wang"
  ],
  "textContent": "**Authors:** Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, Jiaheng Wang\n\nThe 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 [...]",
  "title": "Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?"
}