ϵ-Approximate Coded Matrix Multiplication Is Nearly Twice as Efficient as Exact Multiplication

We study coded distributed matrix multiplication from an approximate recovery viewpoint. We consider a system of
$P$
computation nodes where each node stores
$1/m$
of each multiplicand via linear encoding. Our main result shows that the matrix product can be recovered with
$\epsilon $
relative error from any
$m$
of the
$P$
nodes for any
$\epsilon > 0$
. We obtain this result through a careful specialization of MatDot codes — a class of matrix multiplication codes previously developed in the context of exact recovery (
$\epsilon =0$
). Since prior results showed that MatDot codes achieve the best exact recovery threshold for a class of linear coding schemes, our result shows that allowing for mild approximations leads to a system that is nearly twice as efficient as exact reconstruction. For Entangled-Poly codes — which are generalizations of MatDot codes — we show that approximation reduces the recovery threshold from
$p^{2} q + q -1$
to
$p^{2}q$
, when the input matrices A, B are split respectively in to a
$p \times q$
and
$q \times p$
grids of equal-sized submatrices.