  Spectral algorithms are an important building block in machine learning and
graph algorithms. We are interested in studying when such algorithms can be
applied directly to provide optimal solutions to inference tasks. Previous
works by Abbe, Fan, Wang and Zhong (2020) and by Dhara, Gaudio, Mossel and
Sandon (2022) showed the optimality for community detection in the Stochastic
Block Model (SBM), as well as in a censored variant of the SBM. Here we show
that this optimality is somewhat universal as it carries over to other planted
substructures such as the planted dense subgraph problem and submatrix
localization problem, as well as to a censored version of the planted dense
subgraph problem.
