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Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?

Authors Cornelius Brand , Radu Curticapean , Petteri Kaski , Baitian Li , Ian Orzel , Tim Seppelt , Jiaheng Wang

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LIPIcs.CCC.2026.11.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • arithmetic circuits
  • tensor rank
  • bilinear complexity
  • graph tensors

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Abstract

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 ≥ 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 ≥ 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 ∈ {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 ⊗ 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 (Comput. Complexity 28 (2019) 27-56) and Christandl, Vrana and Zuiddam (Comput. Complexity 28 (2019) 57-111), whose modes correspond to the vertices of undirected graphs H. We make the simple but conceptually crucial observation that Kronecker products T_G ⊗ T_H are isomorphic to T_{G+H}, and that G and H may also be fractional graphs. By asymptotically converting generic tensors to specific graph tensors, we can use nontrivial results from algorithmic graph theory to study the rank and complexity of d-mode tensors for fixed d.

Cite As Get BibTex

Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, and Jiaheng Wang. Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.11

Author Details

Cornelius Brand
  • University of Regensburg, Germany
Radu Curticapean
  • University of Regensburg, Germany
  • IT University of Copenhagen, Denmark
Petteri Kaski
  • Aalto University, Finland
Baitian Li
  • Columbia University, New York, NY, USA
Ian Orzel
  • University of Copenhagen, Denmark
Tim Seppelt
  • IT University of Copenhagen, Denmark
Jiaheng Wang
  • University of Regensburg, Germany
  • University of Helsinki, Finland

Funding

Funded by the European Research Council (ERC) under grant agreement no. 101077083 (CountHom). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency.
  • Li, Baitian: Supported in part by NSF Grant CCF-2238221, a Packard Foundation Fellowship, and a Columbia SEAS Presidential Fellowship.
  • Orzel, Ian: Supported by the ERC under grant agreement no. 101125652 (ALBA).
  • Wang, Jiaheng: Supported by the Helsinki Institute of Information Technology (HIIT).

Acknowledgements

We thank the anonymous referees, whose comments substantially improved the presentation.

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