We aim to understand the extent to which the noise distribution in a planted signal-plus-noise problem impacts its computational complexity. To that end, we consider the planted clique and planted dense subgraph problems, but in a different ambient graph. Instead of Erdős-Rényi $G(n,p)$, which has independent edges, we take the ambient graph to be the random graph with triangles (RGT) obtained by adding triangles to $G(n,p)$. We show that the RGT can be efficiently mapped to the corresponding $G(n,p)$, and moreover, that the planted clique (or dense subgraph) is approximately preserved under this mapping. This constitutes the first average-case reduction transforming dependent noise to independent noise. Together with the easier direction of mapping the ambient graph from Erdős-Rényi to RGT, our results yield a strong equivalence between models. In order to prove our results, we develop a new general framework for reasoning about the validity of average-case reductions based on low sensitivity to perturbations.