Approaching the Chasm at Depth Four

Abstract

Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13] have recently shown that an exp(ω( √ n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √ n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √ n computing the permanent (or the determinant) must be of size exp(Ω( √ n)).

Untitled

“…Lower bounds for this case turn out to be similar to lower bounds for homogeneous depth-4 circuits. In this case we borrow ideas from prior works [13,18,29] and show that the dimension of projected shifted partial derivatives of C is not too large. Most importantly, we can use the chain rule for partial derivatives to obtain good bounds for this complexity measure, independent of the complexity of the various C i .…”

“…Also, in general, it is not clear if these measures are really small for general depth-4 circuits. 13 It is here that the low algebraic rank of {Q i1 , Q i2 , . .…”

“…In the next lemma, we prove an upper bound on the polynomials which are obtained by a composition of low arity polynomials with polynomials of small support. Gupta et al [13] first proved such a bound for homogeneous depth-4 circuit with bounded bottom fan-in.…”

“…Just in the last few years, we have seen rapid progress in proving lower bounds for homogeneous depth-4 arithmetic circuits, starting with the work of Gupta et al [13] who proved exponential lower bounds for homogeneous depth-4 circuits with bounded bottom fan-in and terminating with the results of Kayal et al [18] and of the authors of this paper [29], which showed exponential lower bounds for general homogeneous depth-4 circuits. Any asymptotic improvement in the exponent of these lower bounds would lead to superpolynomial lower bounds for general arithmetic circuits.…”

More Barriers for Rank Methods, via a "numeric to Symbolic" Transfer

“…various forms of circuit and proof complexity. Many important lower bounds in arithmetic complexity are obtained in this fashion, such as the partial derivatives method introduced in Computer Science by [Nis91,NW96b], its generalization, the shifted partial derivatives method -introduced by [Kay12] and developed further in [GKKS14,KS17].…”