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

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 […]

💡 Research Summary

The paper “Beyond Bilinear Complexity: What Works and What Breaks with Many Modes” investigates the computational complexity of tensors with more than three modes, a regime where the well‑known equivalence between asymptotic tensor rank and asymptotic circuit complexity (which holds for bilinear maps) no longer applies. The authors adopt the graph‑tensor framework introduced by Christandl and Zuiddam, where a tensor T_H is associated with an undirected graph H and each vertex of H corresponds to a mode of the tensor. A key observation is that the Kronecker product of two graph tensors satisfies T_G ⊗ T_H ≅ T_{G+H}, where G+H denotes the graph obtained by taking the disjoint union of G and H. This structural identity allows the authors to translate questions about tensor powers and Kronecker products into combinatorial problems on graphs, thereby leveraging a rich toolbox from graph theory.

First, the authors give a new graph‑theoretic proof of Strassen’s classic 2ω/3 bound on the asymptotic rank exponent for three‑mode tensors. Their proof is flexible: instead of fixing the low‑rank auxiliary tensor L to be the matrix‑multiplication tensor (the graph tensor of a triangle K₃), they allow L to be the graph tensor of any complete graph K_s with s ≤ d. By applying this construction to d‑mode tensors, they obtain a universal upper bound of (d − 1)·ω/3 on the asymptotic rank exponent for any fixed d ≥ 3. For d ≥ 4, better bounds are known for the graph tensor of K_d; the authors exploit these by invoking Strassen’s laser method, which yields an improved exponent of 0.772318·(d − 1) for the rank of generic d‑mode tensors. This improves upon the trivial lower bound ⌊d/2⌋ and, assuming the conjectured ω = 2, would give an exponent of 2/3·(d − 1).

Second, the paper turns to asymptotic circuit complexity, denoted η(g). While rank and circuit complexity coincide asymptotically for three‑mode tensors, they diverge for four or more modes. The authors prove that for a generic d‑mode tensor the circuit‑complexity exponent satisfies η ≤ d/2 + 1, and they provide optimized constants for d = 4 and d = 5 using refined graph‑theoretic arguments based on line‑graph treewidth. This shows that, unlike rank, circuit complexity grows essentially linearly with the number of modes.

Third, the authors explore whether the submultiplicativity property of rank under Kronecker products extends to circuit complexity. They show that if circuit complexity were submultiplicative in the sense that C(T ⊗ U) ≤ poly(d, C(T), C(U)) for all d‑mode tensors, then the algebraic complexity class equality VP = VNP would follow. Consequently, under the widely believed separation VP ≠ VNP, such a strong submultiplicativity cannot hold. Moreover, assuming the Hyperclique Conjecture (a fine‑grained hardness hypothesis), the failure of submultiplicativity already appears for tensors with eight modes. This conditional result links the structural property of Kronecker products to deep conjectures in algebraic complexity and fine‑grained complexity.

Nevertheless, a restricted form of submultiplicativity survives for Kronecker powers. By converting generic tensors to specific graph tensors and applying algorithmic results on line‑graph treewidth, the authors obtain upper bounds on the circuit complexity of T^{⊗k} that are essentially multiplicative in k. This demonstrates that while arbitrary Kronecker products may break submultiplicativity, iterated powers retain a controlled growth.

The paper concludes by emphasizing that graph tensors provide a powerful bridge between algebraic complexity and combinatorial graph theory, enabling new upper bounds for both rank and circuit complexity in the multi‑mode setting, and revealing fundamental limits on how these complexities interact under tensor products. The work opens several avenues for future research, including extending the analysis to hypergraph tensors, exploring tighter bounds under alternative graph parameters, and investigating the implications of fine‑grained conjectures such as Hyperclique on practical algorithm design for high‑dimensional multilinear problems.