  We consider a prototypical problem of Bayesian inference for a structured
spiked model: a low-rank signal is corrupted by additive noise. While both
information-theoretic and algorithmic limits are well understood when the noise
is a Gaussian Wigner matrix, the more realistic case of structured noise still
proves to be challenging. To capture the structure while maintaining
mathematical tractability, a line of work has focused on rotationally invariant
noise. However, existing studies either provide sub-optimal algorithms or are
limited to special cases of noise ensembles. In this paper, using tools from
statistical physics (replica method) and random matrix theory (generalized
spherical integrals) we establish the first characterization of the
information-theoretic limits for a noise matrix drawn from a general trace
ensemble. Remarkably, our analysis unveils the asymptotic equivalence between
the rotationally invariant model and a surrogate Gaussian one. Finally, we show
how to saturate the predicted statistical limits using an efficient algorithm
inspired by the theory of adaptive Thouless-Anderson-Palmer (TAP) equations.
