Abstract
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border 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 those that follow from lowrank decompositions. For instance the fastest known O(n^{(omega +epsilon)t}) time algorithms for counting 3tcliques can be implemented with tensor networks, even though the underlying tensor has border rank n^{3t} for all t >= 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.
While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including:
b) an Omega(n^{bw(P)}) time lower bound for counting homomorphisms from P to an nvertex graph, matching the upper bound if omega = 2. In particular for P a vclique this yields an Omega(n^{ceil[2v/3]}) time lower bound for counting vcliques, and for P a kuniform vhyperclique we obtain an Omega(n^v) time lower bound for k >= 3, ruling out tensor networks as an approach to obtaining nontrivial algorithms for hyperclique counting and the Max3CSP problem.
c) an Omega(2^{0.918n}) time lower bound for the permanent of an n x n matrix.
BibTeX  Entry
@InProceedings{austrin_et_al:LIPIcs:2018:10100,
author = {Per Austrin and Petteri Kaski and Kaie Kubjas},
title = {{Tensor Network Complexity of Multilinear Maps}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {7:17:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10100},
URN = {urn:nbn:de:0030drops101006},
doi = {10.4230/LIPIcs.ITCS.2019.7},
annote = {Keywords: arithmetic complexity, lower bound, multilinear map, tensor network}
}
Keywords: 

arithmetic complexity, lower bound, multilinear map, tensor network 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 