Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications


Link to the article

Authors

Suryajith Chillara, Nutan Limaye and Srikanth Srinivasan

Publisher Information

Abstract

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within P. In this paper, we study the algebraic formula complexity of multiplying d many 2×2 matrices, denoted IMMd, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear.

Formally, for each depth Δlogd, we show that any product-depth Δ multilinear formula for IMMd must have size exp(Ω(Δd1/Δ)). It also follows from this that any multilinear circuit of product-depth Δ for the same polynomial of the above form must have a size of exp(Ω(d1/Δ)). In particular, any polynomial-sized multilinear formula for IMMd must have depth Ω(logd), and any polynomial-sized multilinear circuit for IMMd must have depth Ω(logd/loglogd). Both these bounds are tight up to constant factors.

Our lower bound has the following consequences for multilinear formula complexity.

  • Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size s can be converted to one of size sO(1) and depth O(logs); further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to o(logs) without a superpolynomial blow-up in size.

  • Circuits vs. formulas: Any circuit of size s and product-depth Δ can be converted into a formula of product-depth Δ and size sO(Δ). In the multilinear setting, we show that it is not possible to improve on this significantly for small depths. Formally, our results imply that for all large enough s and Δ=o(logs/loglogs), there is an explicit multilinear polynomial Ps,Δ that has a syntactic multilinear circuit of size s and is such that any multilinear formula of product-depth Δ computing Ps,Δ must have size sΩ(Δ).

  • Separations from general formulas: Shpilka and Yehudayoff (FnTTCS 2010) asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a non-trivial upper bound for IMMd implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size s and product-depth Δ=o(logs), general formulas of size s and product-depth Δ cannot be converted to multilinear formulas of size sO(1) and product-depth Δ, when the underlying field has characteristic zero.