  We study the problem of estimating a rank-$1$ signal in the presence of
rotationally invariant noise-a class of perturbations more general than
Gaussian noise. Principal Component Analysis (PCA) provides a natural
estimator, and sharp results on its performance have been obtained in the
high-dimensional regime. Recently, an Approximate Message Passing (AMP)
algorithm has been proposed as an alternative estimator with the potential to
improve the accuracy of PCA. However, the existing analysis of AMP requires an
initialization that is both correlated with the signal and independent of the
noise, which is often unrealistic in practice. In this work, we combine the two
methods, and propose to initialize AMP with PCA. Our main result is a rigorous
asymptotic characterization of the performance of this estimator. Both the AMP
algorithm and its analysis differ from those previously derived in the Gaussian
setting: at every iteration, our AMP algorithm requires a specific term to
account for PCA initialization, while in the Gaussian case, PCA initialization
affects only the first iteration of AMP. The proof is based on a two-phase
artificial AMP that first approximates the PCA estimator and then mimics the
true AMP. Our numerical simulations show an excellent agreement between AMP
results and theoretical predictions, and suggest an interesting open direction
on achieving Bayes-optimal performance.
