Tensor network complexity of multilinear maps

Abstract

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low-rank tensor decompositions, such as $O(n^{\omega +ϵ})$ time matrix multiplication, and in addition many other algorithms such as $O(n\log n)$ time discrete Fourier transform and $O^{\ast }(2^n)$ time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known $O(n^{(\omega +ϵ)t})$ time algorithms for counting $3t$-cliques can be implemented with tensor networks, even though the underlying tensor has rank $n^{3t}$ for all $t\ge 2$. For counting homomorphisms of a general pattern graph $P$ into a host graph on $n$ vertices we obtain an upper bound of $O(n^{(\omega +ϵ)\text{bw}(P)/2})$ where $\text{bw}(P)$ is the branchwidth of $P$. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of $P$. While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including:
(a) an $\mathrm{\Omega }(n^{\text{bw}(P)})$ time lower bound for counting homomorphisms from $P$ to an $n$-vertex graph, matching the upper bound if $\omega =2$. In particular for $P$ a $v$-clique this yields an $\mathrm{\Omega }(n^{\lceil 2v/3\rceil })$ time lower bound for counting $v$-cliques, and for $P$ a $k$-uniform $v$-hyperclique we obtain an $\mathrm{\Omega }(n^v)$ time lower bound for $k\ge 3$, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-$3$-CSP problem.
(b) an $\mathrm{\Omega }(2^{0.918n})$ time lower bound for the permanent of an $n\times n$ matrix.