Seminars

Zoom link: univ-lille-fr.zoom.us/j/95419000064



Titre: Lower bound for arithmetic circuits via the Hankel matrix

Abstract: We study the complexity of representing polynomials by arithmetic
circuits in both the commutative and the non-commutative settings. To
analyse circuits we count their number of parse trees, which describe the
non-associative computations realised by the circuit. In the non-commutative
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^{1/3-ε}} 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 yz).
Our first and main conceptual result is a characterization result: we show
that the size of the smallest circuit computing a given non-associative
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 non-commutative settings, and can be seen
as an extension of the seminal result of Nisan giving a similar
characterization for non-commutative 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.