  We consider a generalization of an important class of high-dimensional
inference problems, namely spiked symmetric matrix models, often used as
probabilistic models for principal component analysis. Such paradigmatic models
have recently attracted a lot of attention from a number of communities due to
their phenomenological richness with statistical-to-computational gaps, while
remaining tractable. We rigorously establish the information-theoretic limits
through the proof of single-letter formulas for the mutual information and
minimum mean-square error. On a technical side we improve the recently
introduced adaptive interpolation method, so that it can be used to study
low-rank models (i.e., estimation problems of "tall matrices") in full
generality, an important step towards the rigorous analysis of more complicated
inference and learning models.
