  The inference of a large symmetric signal-matrix $\mathbf{S} \in
\mathbb{R}^{N\times N}$ corrupted by additive Gaussian noise, is considered for
two regimes of growth of the rank $M$ as a function of $N$. For sub-linear
ranks $M=\Theta(N^\alpha)$ with $\alpha\in(0,1)$ the mutual information and
minimum mean-square error (MMSE) are derived for two classes of
signal-matrices: (a) $\mathbf{S}=\mathbf{X}\mathbf{X}^\intercal$ with entries
of $\mathbf{X}\in\mathbb{R}^{N\times M}$ independent identically distributed;
(b) $\mathbf{S}$ sampled from a rotationally invariant distribution.
Surprisingly, the formulas match the rank-one case. Two efficient algorithms
are explored and conjectured to saturate the MMSE when no
statistical-to-computational gap is present: (1) Decimation Approximate Message
Passing; (2) a spectral algorithm based on a Rotation Invariant Estimator. For
linear ranks $M=\Theta(N)$ the mutual information is rigorously derived for
signal-matrices from a rotationally invariant distribution. Close connections
with scalar inference in free probability are uncovered, which allow to deduce
a simple formula for the MMSE as an integral involving the limiting spectral
measure of the data matrix only. An interesting issue is whether the known
information theoretic phase transitions for rank-one, and hence also
sub-linear-rank, still persist in linear-rank. Our analysis suggests that only
a smoothed-out trace of the transitions persists. Furthermore, the change of
behavior between low and truly high-rank regimes only happens at the linear
scale $\alpha=1$.
