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

2026-02-12

AI Summary

This work addresses the lack of a theoretical characterization of the asymptotic circuit complexity of higher-order tensors (with order $d \geq 4$) and the breakdown of the relationship between tensor rank and complexity in the multilinear setting. It presents the first systematic study of asymptotic circuit complexity for multiway tensors, introducing a novel framework based on graph tensors, refined combinatorial analysis, and the hyperclique conjecture. The main contributions include establishing a general upper bound of $d/2 + 1$ for any order $d$, improving known bounds for $d = 4$ and $d = 5$, and extending Strassen’s trilinear complexity bound to $(d-1)\omega/3$. Furthermore, under fine-grained complexity assumptions, the paper demonstrates that submultiplicativity of the Kronecker product fails for $d = 8$, revealing a fundamental divergence between asymptotic complexity and tensor rank.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $d \geq 4$ 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 $2\omega/3$ bound on the asymptotic rank exponent of $3$-mode tensors. Our proof directly generalizes to an upper bound of $(d-1)\omega/3$ 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 $\omega$. 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 [...]

Problem

• tensor complexity
• circuit complexity
• multilinear maps
• asymptotic rank
• fine-grained complexity

Innovation

• tensor complexity
• circuit complexity
• submultiplicativity
• graph tensors
• fine-grained complexity