Lower Bounds For Depth 4 Formulas Computing Iterated Matrix Multiplication

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 x n, IMM,d, has size ric-(`R) as long as d = n (1). This improves the result of Nisan and Wigderson (Computational Complexity, 1997) for depth 4 set-multilinear formulas. We also study E11[ (dit)]Ell[t] formulas, which are depth 4 formulas with the stated bounds on the fan-ins of the H gates. A recent depth reduction result of Tavenas (MFCS, 2013) shows that any n-variate degree d = n (1) polynomial computable by a circuit of size poly(n) can also be computed by a depth 4 E11[0(dit)]E11[t] formula of top fan-in n (dit). We show that any such formula computing IMM,d has top fan-in rtc4dit), 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.