The new randomized transform can compute the sketched pair (SA, Sb) in O(nd + ε^-2 d^4) time using one Hadamard pass

uses one Hadamard pass with a padded internal dimension $N=\widetilde{O}(n+ε^{-2}d^3)$, and is efficient to apply: the sketched pair $(SA, Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+ε^{-2}d^4)$ time.

http://arxiv.org/abs/2608.26552v1