Remark: 135-01. Homogeneous depth-five lower bounds for iterated matrix multiplication

Authors: OpenAI. Family 135.

Title: Homogeneous depth-five lower bounds for iterated matrix multiplication.

Abstract: Let ${}mathop{{}mathrm{IMM}}{}nolimits _{n,n}$ be the $(1,1)$ entry of a product of n independent $n{}times n$ matrices of variables. Over every field of characteristic zero, every syntactically homogeneous ${}Sigma{}Pi{}Sigma{}Pi{}Sigma$ circuit computing ${}mathop{{}mathrm{IMM}}{}nolimits _{n,n}$ has at least $n{}{{}sqrt n/400}$ gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most $n{}{{}sqrt n+4}$ gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is $n{}{{}Theta({}sqrt n)}$.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.