Abstract
We study the complexity of representing polynomials by arithmetic circuits in both the commutative and the noncommutative settings. To analyse circuits we count their number of parse trees, which describe the nonassociative computations realised by the circuit.
In the noncommutative setting a circuit computing a polynomial of degree d has at most 2^{O(d)} parse trees. Previous superpolynomial lower bounds were known for circuits with up to 2^{d^{1/3ε}} parse trees, for any ε > 0. Our main result is to reduce the gap by showing a superpolynomial lower bound for circuits with just a small defect in the exponent for the total number of parse trees, that is 2^{d^{1  ε}}, for any ε > 0.
In the commutative setting a circuit computing a polynomial of degree d has at most 2^{O(d log d)} parse trees. We show a superpolynomial lower bound for circuits with up to 2^{d^{1/3  ε}} parse trees, for any ε > 0. When d is polylogarithmic in n, we push this further to up to 2^{d^{1  ε}} parse trees.
While these two main results hold in the associative setting, our approach goes through a precise understanding of the more restricted setting where multiplication is not associative, meaning that we distinguish the polynomials (xy)z and x(yz). Our first and main conceptual result is a characterization result: we show that the size of the smallest circuit computing a given nonassociative polynomial is exactly the rank of a matrix constructed from the polynomial and called the Hankel matrix. This result applies to the class of all circuits in both commutative and noncommutative settings, and can be seen as an extension of the seminal result of Nisan giving a similar characterization for noncommutative algebraic branching programs. Our key technical contribution is to provide generic lower bound theorems based on analyzing and decomposing the Hankel matrix, from which we derive the results mentioned above.
The study of the Hankel matrix also provides a unifying approach for proving lower bounds for polynomials in the (classical) associative setting. We demonstrate this by giving alternative proofs of recent lower bounds as corollaries of our generic lower bound results.
BibTeX  Entry
@InProceedings{fijalkow_et_al:LIPIcs:2020:11885,
author = {Nathana{\"e}l Fijalkow and Guillaume Lagarde and Pierre Ohlmann and Olivier Serre},
title = {{Lower Bounds for Arithmetic Circuits via the Hankel Matrix}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {24:124:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771405},
ISSN = {18688969},
year = {2020},
volume = {154},
editor = {Christophe Paul and Markus Bl{\"a}ser},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/11885},
URN = {urn:nbn:de:0030drops118859},
doi = {10.4230/LIPIcs.STACS.2020.24},
annote = {Keywords: Arithmetic Circuit Complexity, Lower Bounds, Parse Trees, Hankel Matrix}
}
Keywords: 

Arithmetic Circuit Complexity, Lower Bounds, Parse Trees, Hankel Matrix 
Collection: 

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020) 
Issue Date: 

2020 
Date of publication: 

04.03.2020 