Lower bounds for depth 4 formulas computing iterated matrix multiplication

Abstract

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth 4 arithmetic formula computing the product of d generic matrices of size n × n, IMMn,d, has size nΩ(√d) as long as d = nO(1). This improves the result of Nisan and Wigderson (Computational Complexity, 1997) for depth 4 set-multilinear formulas. We also study ΣΠ[O(d/t)] ΣΠ[t] formulas, which are depth 4 formulas with the stated bounds on the fan-ins of the Π gates. A recent depth reduction result of Tavenas (MFCS, 2013) shows that any n-variate degree d = nO(1) polynomial computable by a circuit of size poly(n) can also be computed by a depth 4 ΣΠ[O(d/t)] ΣΠ[t] formula of top fan-in nO(d/t). We show that any such formula computing IMMn,d has top fan-in nΩ(d/t), proving the optimality of Tavenas' result. This also strengthens a result of Kayal, Saha, and Saptharishi (ECCC, 2013) which gives a similar lower bound for an explicit polynomial in VNP.