I have matrices $A$ and $B$ and vector $x$, and am interested in computing $|AB|x$ efficiently, where $||$ is element-wise absolute value. Is there any neat equality I can use to avoid computing $|AB|$?
E.g. say $A$ is $100 \times 1000$, $B$ is $1000 \times 100$ and $x$ is $100$-D. If I wanted to compute $ABx$ the fastest way would be to do $A(Bx)$ while doing $(AB)x$ would be roughly $100\times$ slower. However, I can't do that for my problem of computing $|AB|x$ as it seems I have to compute $AB$ first.
If it helps: all elements of $x$ are non-negative, there are no constraints on $A$ and $B$.