On the Power of Homogeneous Depth 4 Arithmetic Circuits

Abstract

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in VP. Our results hold for the Iterated Matrix Multiplication polynomial -in particular we show that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension n O(1) must have size n Ω( √ n) . Our results strengthen previous works in two significant ways.1. Our lower bounds hold for a polynomial in VP. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in VNP. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in VP was the bound of n Ω(log n) by [LSS, KLSS14].Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin).2. Our lower bound holds over all fields. The lower bound of [KLSS14] worked only over fields of characteristic zero. Prior to our work, the best lower bound for homogeneous depth 4 circuits over fields of positive characteristic was n Ω(log n) [LSS,KLSS14].

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“…For, say, d ∈ o(lg N/ lg lg N), and for sufficiently large N, any α > 0 works.) All previous lower bound results using the shifted partial derivative method also obtain similar statements [9,6,17,16].…”

“…Our analysis is quite different from those in previous papers (such as [6,12,16]), which are based on either monomial counting (meaning that we find a large identity or upper triangular submatrix inside our matrix) or an analytic inequality of Alon [2]. Neither of these techniques seems to be applicable in our case.…”

“…Later, a similar result for a polynomial in VP was proved in [6] and this was subsequently strengthened by Kumar and Saraf [17], who gave a polynomial computable by homogeneous ΠΣΠ circuits such that any ΣΠ [O(d/t)] ΣΠ [t] circuits computing it must have top fan-in N Ω(d/t) . Finally, using a variant of the shifted partial derivative measure, Kayal et al [12] and Kumar and Saraf [16] were able to prove similar lower bounds for general depth-4 homogeneous circuits as well.…”

Recent Progress on Lower Bounds for Arithmetic Circuits

“…If we could do this over the reals/complex numbers, then given the [KLSS14] result, this would also suffice in separating VP from VNP! Very recently, Kumar and Saraf [KS14] built upon and extended the results of [KLSS14] to hold over all fields. They achieved this via giving a new and more combinatorial proof of the result of [KLSS14] that is not dependent on the underlying field.…”

“…Theorem 3.8 ( [KS14]): Let F be any field. There exists an explicit family of polynomials (over F) of degree n and in N = n O(1) variables in VP, such that any homogeneous ΣΠΣΠ circuit computing it has size at least n Ω( √ n) .…”