Tensor Network Complexity of Multilinear Maps

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 $O(n^{\omega+\epsilon})$ time matrix multiplication, and in addition many other algorithms such as $O(n \log n)$ time discrete Fourier transform and $O^*(2^n)$ time for computing the permanent of a matrix. However, tensor networks sometimes yield faster algorithms than thosethat follow from low-rank decompositions. For instance the fastest known $O(n^{(\omega +\epsilon)t})$ 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 $O(n^{(\omega+\epsilon)\bw(P)/2})$ where $\bw(P)$ is the branchwidth of $P$. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of $P$.