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

TL;DR

It is shown that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes, and can salvage a restricted notion of submultiplicativity for the circuit complexity of tensors with many modes.

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 $2ω/3$ 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 [...]

Figures (8)

• Figure 1: Decomposition of $2\cdot K_d$ into $d$ stars, \ref{['eq:star-clique']}, for $d=4$.
• Figure 2: The grid $G$ is the sum of four matchings, so \ref{['lem:decomp']} shows that its graph tensor $T_{G,N}$ is the Kronecker product of four graph tensors of matchings. But since grids are hard by \ref{['lem:per-from-grid']} and matchings are easy by \ref{['ex:matching']}, we do not expect complexity to be submultiplicative.
• Figure 3: The reduction from the permanent to the graph tensors of grids, as shown in \ref{['lem:per-from-grid']}. The gray and orange edges are assigned $0$ and $1$, respectively. Each horizontal and each vertical path flips from $0$ to $1$ exactly once; whenever this happens at some vertex $v_{i,j}$ (a flip vertex, shown purple), we ensure that both the $i$-th horizontal and the $j$-th vertical path flip at $v_{i,j}$. In other words, flip vertices always lie on orange corners. Each flip vertex $v_{i,j}$ contributes $x_{i,j}$ to the total weight of the assignment, otherwise $1$. The weight of the assignment shown here is $x_{11}x_{23}x_{34}x_{42}$, which is indeed of the form $\prod_i x_{i,\pi(i)}$ for $\pi =1342$.
• Figure 4: A non-border signature. Below an assignment is the value of the signature. Assignments that are not listed here evaluate to zero.
• Figure 5: Decomposition of the incidence graph $I$ of the $(3,4)$-hyperclique into matchings ${\color{cbfp1} M_1},{\color{cbfp2} M_2},{\color{cbfp3} M_3}$, where $\underline{1},\underline{2},\underline{3},\underline{4}$ indicate the hyperedges $234,134,124 ,123$, respectively.

Theorems & Definitions (94)

• Remark
• Theorem 0: Upper bound on asymptotic rank
• Theorem 0: Submultiplicativity of $\CC$ implies $\VP = \VNP$
• Theorem 0: Submultiplicativity of $\CC$ implies faster permanents
• Theorem 0: Submultiplicativity of $\CC$ implies faster hypercliques
• Theorem 0: Mixed asymptotic rank and circuit complexity
• Theorem 1: Upper bound on asymptotic circuit complexity
• Theorem 2: Upper bound on asymptotic circuit complexity for small orders
• Theorem 3: Yates's algorithm
• Definition 4: Graph tensor